REESE  LIBRARY 

OF  THE 

UNIVERSITY  OF  CALIFORNIA. 


,:;...,,,.::.v.,,^9o-:r        f 

^Accession  No.  .92342 •  Class  A': 


WORKS  OF   J.  H.  CROMWELL 


PUBLISHED    BY 


JOHN   WILEY   &   SONS. 


A  Treatise  on  Toothed  Gearing:. 

1-2010,  cloth,  $1.50 

A  Treatise  on  Belts  and  Pulleys. 

12010,  cloth,  $1.50. 


A  TREATISE 


ON 


TOOTHED   GEARING. 

Containing  Complete  Enstructions 

FOR  DESIGNING,  DRAWING,  AND  CONSTRUCTING 

SPUR   WHEELS,   BEVEL   WHEELS,   LANTERN 

GEAR,   SCREW   GEAR,   WORMS,    ETC., 


THE  PROPER  FORMATION  OF  TOOTH-PROFILES. 

FOR    THE    USE   OF 

MACHINISTS,   PATTERN-MAKERS,   DRAUGHTSMEN, 
DESIGNERS,   SCIENTIFIC  SCHOOLS,   ETC. 


BY 

J.  HOWARD   CROMWELL,  Pn.B. 

FOURTH    EDITION. 
SECOND   THOUSAND. 

NEW    YORK: 

JOHN    WILEY    AND    SONS, 

43-45  EAST  NINETEENTH  STREET. 
1901. 


COPYRIGHT,  1883, 
BY  J.  HOWARD  CROMWELL. 


PREFACE. 


IN  presenting  to  the  mechanical  public  this  little  work,  I  am 
fully  aware  that  I  am  treading  upon  well-worn  ground,  and  that 
I  have  devoted  time  and  labor  to  a  subject  which  is  well-nigh 
"old  as  the  hills,"  and  likewise,  to  many,  as  familiar.  It  may 
also  seem  to  some,  who  have  read  more  extensively  than  I  have 
upon  the  subject  of  toothed  gearing,  that  this  book  contains 
nothing  new,  or  original  with  its  author :  had  such  been  my 
belief,  the  book  would  never  have  been  written,  much  less 
published. 

In  my  experience  as  a  mechanical  engineer  I  have  sought 
often  and  earnestly,  but  always  in  vain,  for  a  terse,  compact, 
yet  complete  and  comprehensive  work  on  the  subject  of  toothed 
gearing.  Compelled,  therefore,  by  necessity  to  gain  the  requi- 
site knowledge  from  many  works,  and  also  from  some  failures 
on  my  own  part,  and  believing,  that,  in  the  crowded  field  of 
technical  literature,  room  yet  remained  for  such  a  publication. 
I  decided  to  write  a  book  on  toothed  gearing,  which  should 
contain  all  that  I  had  dug  out  from  so  many  sources,  and  ns 
much  more  as  my  experience  and  originality  had  taught  me, 
yet  being  concise,  terse,  and  simple  enough  to  suit  even  "  the 
wayfaring  man,  though  a  fool."  Such  were  the  somewhat 

iii 

92342 


IV  PREFACE. 

exalted  intentions  of  the  author  in  writing  this  book  :  whether  or 
not  the  reality  equals  the  anticipation,  is  for  the  reader  to  judge. 

Notwithstanding  the  apparent  tendency  to  lay  aside  the  old 
and  simple  "  rules  of  thumb  "  for  the  surer  and  better  methods, 
involving,  to  a  certain  extent,  a  knowledge  of  algebra  and  geom- 
etry, there  are  still  many  mechanics  who  continue  to  look  with 
extreme  distrust  upon  any  thing  in  the  shape  of  a  book,  because 
"  books  are  generally  too  deep  and  too  theoretical."  For  this 
reason  I  have  given  throughout  the  following  pages  simple  rules, 
as  well  as  formulas,  for  performing  each  and  every  operation 
necessary  in  designing  and  laying  out  the  various  kinds  of  gears. 

He  who  possesses  the  requisite  knowledge  of  algebra  and 
geometry  —  for  which  any  man  will  be  the  better  off —  may 
make  use  of  the  formulas  in  designing  the  gears  he  may  have 
to  construct ;  while  he  whose  knowledge  of  mathematics  goes 
not  beyond  the  simple  rules  of  arithmetic  may  obtain  precisely 
the  same  results,  and  do  in  every  way  as  good  work,  by  using  the 
corresponding  rules.  Throughout  the  book  I  have  used  a  uni- 
form system  of  notation  in  order  to  avoid  confusing  or  burden- 
ing the  memory  of  the  reader,  and  the  numerous  examples  will 
serve  to  illustrate  sufficiently  the  application  of  the  various  rules 
and  formulas.  In  all  cases  where  the  contrary  is  not  stated, 
forces  and  weights  are  taken  in  pounds,  and  dimensions  in 
inches.  I  have  also  carefully  avoided  any  use  of  the  metric 
system  ;  because  I  believe  the  good  old  English  inch,  foot,  and 
pound  to  be  accurate  enough  for  the  proper  construction  of 
any  machine,  engine,  or  thing  which  can  be  made  by  the  use 
of  the  metric  system.  In  fact,  American  and  English  machin- 
ery being  the  best  in  the  world,  I  see  no  reason  to  doubt  the 
efficacy  of  the  English  system  of  weights  and  measures,  from 
a  machinal  point  of  view  at  least.  In  writing  upon  a  subject 


PREFACE.  V 

so  old,  and  upon  which  so  much  has  been  written  from  time 
to  time,  it  is  impossible  that  I  should  not,  to  a  certain  extent, 
have  copied  the  thoughts  of  others,  even  though  in  many  cases 
they  are  also  honestly  my  own.  I  deem  it  best,  therefore,  to  say 
that  I  have  taken  the  liberty  of  referring  to  and  quoting  such 
standard  writers  as  Reuleaux,  Camus,  Unwin,  Haswell,  and 
others,  but  never,  I  believe,  without  giving  them  due  credit. 
In  writing  the  paragraph  on  "  Special  Applications  of  the  Prin- 
ciples of  Toothed  Gearing,"  I  have  been  greatly  assisted  by 
referring  to  Mr.  Henry  T.  Brown's  valuable  little  book  entitled 
"  507  Mechanical  Movements,"  without  which  the  work  of  col- 
lecting the  various  contrivances  explained  in  this  paragraph 
would  have  been  indeed  laborious.  I  trust,  that,  while  much 
that  is  printed  in  this  book  may  be  found  in  other  works  on 
the  subject,  it  also  contains  much  that  cannot  be  found  else- 
where, and  that  my  earnest  desire  to  make  it  a  simple,  compre- 
hensive, and  convenient  companion  in  the  shop  and  scientific 
school,  may  be  in  some  measure,  if  not  fully,  realized. 

J.  H.  C 

NEW  YORK,  Feb.  i,  1884. 


TABLE   OF  CONTENTS. 


SECTION   I. 

PAGE. 
Introduction.  —  Fundamental  Principles.  — The  First  Gear-Wheel. 

—  First  Transformation i 


SECTION    II. 

Proper  Form  of  Tooth- Profiles.  —  The  Epicycloid  and  Hypocycloid. 
—  Conditions  necessary  for  Minimum  Friction.  —  Conditions 
necessary  for  Uniform  Velocity.  —  Proper  Size  of  Generating 
Circle.  —  The  Involute 9 


SECTION    III. 

Comparison.  —  Advantages  and  Disadvantages  of  Cycloidal  and 
Involute  Teeth.  —  Experiments  with  Involute  Teeth.  —  The  In- 
volute a  Limiting  Case  of  the  Epicycloid 


SECTION    IV. 

Practical  Methods  for  laying  out  Teeth,  Exact  and  Approximate. 
—  Epicycloidal  Faces  and  Hypocycloidal  Flanks.  —  Involute 
Teeth.  — Straight  Flanks 28 


SECTION    V. 

Rack.  —  Internal  Gears.  —  Methods  for  laying  out  their  Teeth         ..       37 

vii 


viii  CONTENTS. 

SECTION   VI. 

PAGE. 
Special    Forms.  —  External  and   Internal    Lantern  Gears.  —  Mixed 

Gears.  —  Gear  at  Two  Points       .......      42 

SECTION    VII. 

Bevel  Gears.  —  Pitch  Cones.  —  Supplementary  Cones.  —  Method  for 
laying  out  the  Teeth.  —  Internal  Bevels.  —  The  Disk  or  Plane 
Wheel  ........  ....  49 

SECTION   VIII. 

Screw  Gears.  —  Angles  of  the  Teeth  and  Shafts.  —  Screw  Gear  and 
Spur  Pinion.  —  Screw  Rack  and  Pinion.  —  Method  for  laying 
out  the  Teeth.  —  Worm  and  Wheel.  —  Worm  and  Rack.  —  In- 
ternal Worm  Wheel  .........  54 

SECTION    IX. 

Hyperbolic  Gears.  —  Calculations.  —  Examples.  —  Teeth  of  Hyper- 

bolic Gears    ...........      65 

SECTION   X. 

Relations  between  Diameter,  Circumference,  Pitch,  Number  of 
Teeth,  etc.  —  Diametral  Pitch.  —  Methods  for  stepping  off  the 
Pitch  on  the  Pitch  Circle  ........  72 

SECTION   XI. 
Ratios.  —  Velocity.  —  Revolution.  —  Power.  —  Examples  ...      78 


SECTION    XII. 

Line  of  Contact.  —  Arcs  of  Approach  and  Recess.  —  Arc  of  Con- 
tact       .  .......... 


.    CONTENTS.  IX 

SECTION    XIII. 

PAGF.. 
Strength  of  Teeth.  —  Rules  and  Formulas  for  determining  the  Pitch 

and  Other  Tooth  Dimensions.  —  Tables  for  determining  the 
Pitch.  —  Examples.  —  Table  for  converting  Decimals  into  Frac- 
tions. —  High  Speed  Gears 89 

SECTION    XIV. 

Strength  of  Arms.  —  Rectangular,  Circular,  Elliptical,  and  Flanged 
Cross-Sections.  —  Number  of  Arms.  —  Rim,  Nave,  Shafts,  etc. 
—  Tables  for  determining  Diameters  of  Steel  and  Wrought- 
Iron  Shafts. —  Approximate  Weight  of  Gear-Wheels  .  .  107 


SECTION    XV. 
Recapitulation  of  Formulas  and  Rules,  with  Uniform  Notation        .     139 

SECTION   XVI. 

Complete  Design  of  Spur  Wheel,  Bevel  Wheels,  Screw  Gears, 
Worm  and  Wheel,  Internal  Gears,  Lantern  Gears,  and  Gear 
Train,  with  Full  Working  Drawings 151 

SECTION    XVII. 

Special  Applications  of  the  Principles  of  Toothed  Gearing. — 
Devices  for  producing  Variable  Motion.  —  Rectangular  Gears. 
—  Triangular  Gears.  —  Elliptical  Gears.  —  Scroll  Gears,  etc.  .  197 

APPENDIX. 

Relative  Values  of  Circumferential  and  Diametral  Pitches. —  Ex- 
planation of  the  Process  of  cutting  Gear-Teeth.  —  Diametral 
Rules  and  Formulas -^ 3 


TOOTHED  GEARING. 


§  I.  —  Introduction.  —  Fundamental  Principles. 

IN  the  Science  of  Machinery,  a  science  of  vast  conse- 
quence to  the  world,  and  vital  to  the  wealth  and  power 
of  any  nation,  there  is,  perhaps,  no  more  important 
branch  than  the  transmission  of  power  and  motion  by 
means  of  toothed  gearing ;  for  in  toothed  gearing  we 
have  practically  the  only  means  of  the  all-necessary 
transmission.  Having  been  known  for  thousands  of 
years,  and  in  practical  use  for  centuries,  in  reviewing 
this  subject  we  should  naturally  look  for  many  succes- 
sive alterations  and  improvements,  even  in  fundamental 
principle ;  but  no  such  result  will  be  found  by  the  most 
diligent  research.  Contrary  to  the  natural  and  seem- 
ingly inevitable  course  of  mechanical  contrivances,  in 
principle  toothed  gearing  stands  as  an  exception  to  the 
well-nigh  universally  accepted  theory  of  "small  begin- 
ning and  gradual  development."  Improvement  in  this 
branch  of  machinal  science  has  been  slow  and  retarded ; 
and  strangely  discordant  with  the  general  belief  that 
first  principles  are  always  erroneous,  or  at  least  faulty 
ones,  is  the  fact  that  the  fundamental  principle  of 
toothed  gearing,  as  it  may  be  expressed  to-day,  is  pre- 


TOOTHED    GEARING. 


cisely  what  it  was  ten  centuries  ago.  The  slow-moving 
centuries  which  have  witnessed  the  successive  changes 
in  water-motors  —  from  the  simple  undershot  wheel, 
driven  in  mid-stream  by  the  impulsive  force  of  the 
river's  current,  first  to  the  overshot  and  Poncelet,  then 
to  the  turbine  and  water-engine  of  the  nineteenth  cen- 
tury, each  involving  a  different,  and,  in  its  turn,  an 
improved,  principle  —  can  tell  of  no  such  advance  in 
the  essential  principle  of  toothed  gearing.  Throughout 
the  years  which  have  changed  the  steam-engine  from 
an  atmospheric-pressure  engine  to  a  high-pressure  ex- 
pansion steam-motor  ;  throughout  the  years  which  have 
produced  the  locomotive-engine,  the  ocean  steamer,  the 
telegraph,  the  electric  light,  the  gas-engine,  and  the 
telephone,  with  all  their  successive  alterations  in  prin- 


Fig.  I 

/* 

& 

B 


A     (£ 


ciple  and  theory, — the  science  of  toothed  gearing  almost 
alone  has  been  able  to  attest,  that  in  one  case  at  least, 
if  no  more,  first  principles  have  been  sound  and  per- 
fect, —  so  perfect  as  to  stand  the  test  of  years  without 
change  or  improvement.  This  principle,  most  simple, 
although  the  underlying  principle  of  the  whole  theory 
and  study  of  toothed  gearing,  may  be  succinctly  ex- 


TOOTHED   GEARING.  3 

pressed  as  follows :  If  two  cross-shaped  pieces  be  placed 
as  in  Fig.  i,  the  arms  of  A  being  somewhat  shorter  than 
those  of  B,  and  the  pieces  being  allowed  only  the  motion 
of  rotation  about  their  fixed  axes,  or  centres,  then,  if  a 
continuous  rotary  motion  in  the  direction  indicated  by 
the  arrow  be  given  to  the  piece  B,  a  similarly  contin- 
uous rotary  motion  in  the  opposite  direction  will  be 
given  to  the  piece  A.  For  the  arm  a,  in  contact  with 
the  arm  a1 ,  will  act  as  a  lever  upon  it,  forcing  it  down- 
ward, and  at  the  same  time  bringing  the  arms  b  and  b' 
into  such  relative  positions,  that  a  similar  action  will 
take  place  between  them.  Thus  successively  each  arm 
of  the  piece  B  will  act  upon  the  corresponding  arm  of 
the  piece  A,  and  a  continuous  rotary  motion  will  be 
transmitted  from  the  piece  B  to  the  piece  A.  Simple 


Fig.2 


and  crude  as  our  sketch  may  appear,  and  however 
childish  and  primary  our  statement  of  this  fundamental 
principle  may  seem,  a  most  complete  analogy  exists 
between  them  and  the  most  smoothly  and  accurately 
running  gears  of  the  present  day ;  for  each  one  of  the 
countless  scores  of  accurately  profiled  teeth,  working 
so  industriously  and  almost  noiselessly  in  our  machine- 


TOOTHED   GEARING. 


shops  and  factories,  is  but  the  projecting  arm  of  our 
cross-shaped  pieces,  modified  in  accordance  with  the 
advance  in  machine  manufacture,  and  shaped  to  suit 
the  increased  demand  for  accuracy  of  transmission. 
Since,  doubtless,  the  first  gear-wheels  were  similar  to 
those  represented  in  our  figure,  let  us  examine  a  little 
more  minutely  their  action  and  the  conditions  neces- 
sary for  such  action.  Let  us  suppose  each  wheel  to 
consist  of  three  long,  slender  pieces,  or  arms,  crossed 
and  fixed  in  such  a  manner  that  their  ends  divide  the 
circumscribing  circles  into  six  equal  arcs  ;  that  is,  they 
form  the  diagonals  of  a  regular  hexagon  (Fig.  2).  The 
arrows  indicate  the  directions  in  which  the  wheels 
revolve.  Now,  in  order  that  the  rotary  motion  be 
continuous,  it  is  obvious  that  contact  between  the  arms 


d  and  d'  must  not  cease  until  contact  is  begun  between 
the  following  pair  of  arms,  c  and  c:  otherwise  the 
wheel  o  would  move  some  distance  without  moving  the 
wheel  </,  and  consequently  the  motion  of  the  wheel  o 
would  be  intermittent.  It  is  also  necessary  that  the 
arms  of  of  be  somewhat  shorter  than  those  of  o;  for  if 
they  were  equal  (Fig.  3),  the  arcs  xqy  and  xcfy  being 
also  equal,  the  arms  a  and  a  would  come  in  contact  at 


TOOTHED   GEARING. 


5 


their  ends,  and  rotation  would  be  impossible,  or,  for  a 
greater  separation,  the  arm  /;  would  leave  the  arm  // 
before  the  arms  a  and  af  had  reached  their  proper  posi- 
tions, and  the  wheel  o  would  move  on  indefinitely  with- 
out touching  the  wheel  o1  (Fig.  4). 


Fig.4 


A  glance  at  Fig.  5  is  sufficient  to  show  the  impos- 
sibility of  continuous  transmission  from  o  to  o1  when 
the  arms  of  or  are  longer  than  those  of  o.  Let  r  be  the 
(ength  of  each  arm  of  the  wheel  o  (Fig.  2),  and  / 


the  length  of  each  arm  of  the  wheel  o'.  Let  us  sup- 
pose that  contact  between  the  arms  c  and  c'  begins  at 
the  moment  when  contact  between  d  and  dr  is  just 
about  to  cease.  We  have  then  the  distance  pR,  be- 
tween the  two  points  of  contact  equal  to  /,  because 


6  TOOTHED   GEARING. 

opR  is  an  equilateral  triangle.  But  we  have  seen  that 
r'  must  be  less  than  r:  consequently  pR  must  be  less 
than  r.  The  distance  pR  must  obviously  be  greater 
than  the  distance  pS,  else  there  would  be  no  contact  at 
all  between  the  arms  c  and  c' .  Since,  now,  the  line/  W 
is  perpendicular  to  and  bisects  the  arm  c,  we  have 


==\/x;2 


2  =  y/f r2  =  .866r, 


but  pR  =  r'  is  greater  than  pS:  hence  the  'conditions 
necessary  for  uniform  transmission  from  the  wheel  o  to 
the  wheel  o'  are,  that  r'  be  less  than  r  and  greater  than 
.866r.  If  there  were  a  wheel  of  this  sort  given,  to  be 
used  as  a  driver,  and  we  wished  to  construct  a  wheel 


Fig. 6 


which  would  gear  continuously  with  it,  we  would  pro- 
ceed  as  follows:  From  the  point /,  with  a  radius  less 
than  r  and  greater  than  .866r,  say  .gr,  we  describe  an 
arc  cutting  the  arm  c  in  some  point  R.  Then,  with 
the  same  radius,  we  describe  a  circle  passing  through  the 
points  R  and/,  and  draw  the  diagonals  of  the  regular 
inscribed  hexagon,  of  which  pR  is  one  side.  The  end 
of  the  arm  o'c  (Fig.  6)  comes  in  contact  with  the  arm 
oe  at  the  point  r,  and  slides  along  its  surface  until  the 


TOOTHED   GEARING.  J 

arms  have  assumed  the  positions  o'g  and  of  respective- 
ly. Then  the  end  of  the  arm  oe  (now  in  the  position 
of}  comes  in  contact  with  the  arm  o'c  (now  o'g)  at  the 
point  f,  and  slides  along  its  surface  until  the  positions 
ob  and  o'b  are  reached,  after  which  contact  between  this 
pair  of  arms  ceases.  That  is,  during  each  revolution 
the  end  of  the  arm  o'c  rubs  along  the  surface  of  the  arm 
oe  for  the  distance  cd,  and  the  end  of  oe  rubs  along  the 
surface  of  o'c  for  the  greater  distance  ab.  The  wearing- 
surfaces  being  unequal  in  the  two  wheels,  the  wear  will 
be  unequal,  or,  in  other  words,  one  wheel  will  wear  out 
before  the  other :  thus  the  accuracy  of  transmission 
will  soon  be  destroyed,  and  the  wheels  rendered  useless. 
Such  rude  contrivances  can,  of  course,  be  of  no  practical 
use,  and  are  given  here,  not  as  practical  examples,  but 
because  of  their  natural  primitiveness,  and  because  they 
embody  principles  from  which  has  been  built  up  the 
present  complete  theory  of  toothed  gearing.  Whether 
or  not  these  primitive  gear-wheels  were  ever  used  for 
actual  transmission,  is  indeed  uncertain ;  and  aside  from 
the  natural  conclusion  that  the  science  of  toothed  gear- 
ing, like  all  other  sciences,  must  have  sprung  from  a 
mere  germinal  conception,  and  that  our  simple  crossed 
pieces  were  most  probably  the  first  tangible  form,  the 
evidence  of  their  real  existence  is  confined  to  a  few 
rough  old  drawings,  such  as  those  representing  the 
ancient  Greek  and  Asiatic  norias  for  hoisting  water,  in 
which  crossed  pieces  of  wood  precisely  similar  to  our 
Fig.  i  are  delineated.  Certain  it  is,  however,  that  if 
these  crossed  pieces  were  ever  in  actual  use,  time  soon 
effaced  the  crudeness  of  their  construction,  and  obliter- 
ated the  faults  which  caused  their  inutility.  The  num- 


8  TOOTHED   GEARING. 

her  of  arms  was  greatly  increased  ;  the  arms  themselves 
changed  into  pegs,  or  teeth,  projecting  at  regular  inter- 
vals from  the  circumferences  of  drums  or  wheels,  and 
formed  with  curved  profiles,  in  order  to  distribute  the 
wear  evenly  over  the  whole  surfaces  of  the  teeth,  and, 
if  possible,  to  diminish  the  friction  between  the  teeth, 
and  so  also  the  wear  itself  (Fig.  7).  Since,  now,  the  teeth 

Fig.7 


of  the  wheels  A  and  B  rub  or  slide  against  each  other 
when  in  contact,  and  thus  produce  friction  and  wear, 
there  must  be  some  form  of  profile,  straight  or  curved, 
simple  or  compound,  which,  when  given  to  the  teeth, 
will  reduce  the  friction  between  them  to  a  minimum,  — 
some  form  which  will  be  more  advantageous  for  accurate 
transmission  and  uniformity  of  motion  than  any  other. 
It  is  needless  here  to  state  of  what  vast  importance  is 
this  desired  form  of  tooth-profile  ;  for  the  perfection  of 
almost  every  machine  —  the  most  simple  and  compact,  as 
well  as  the  most  complicated  and  extensive  —  depends, 
to  a  very  great  degree,  upon  the  action  of  its  gear- 
wheels, and  consequently  upon  the  formation  of  the 
tooth-profiles  of  the  gears.  The  proper  formation  of 


TOOTHED   GEARING.  9 

the  tooth-profile  must  insure,  in  the  words  of  another, 
"a  more  equable  performance  of  the  work  in  hand,  a 
diminution  of  the  moving-power  wasted  by  friction,  and 
hence  the  accomplishment  of  more  work  with  the  same 
amount  of  power,  and  a  greater  durability,  and  conse- 
quently a  less  cost  for  repairs  in  the  whole  machine." 
Recognizing,  then,  the  fact  that  the  subject  with  which 
we  are  dealing  is  of  more  than  ordinary  importance,  we 
propose  an  investigation  which  aims  to  present,  in  as 
clear  and  terse  a  manner  as  possible,  the  method  of 
reasoning  by  which  the  present  development  of  tooth- 
profiles  has  been  attained,  —  an  investigation  from 
which  have  been  purposely  omitted  all  the  more  intri- 
cate and  tedious  mathematical  calculations  pertaining 
to  the  subject,  which  have  been  so  laboriously  worked 
out  by  other  writers  and  investigators.  Far  from  think- 
ing, or  even  wishing,  to  disparage  the  labors  of  men  of 
genius  and  ability  who  have  devoted  their  time  and 
energies  to  the  promotion  of  the  purely  mathematical 
and  theoretical  part  of  the  great  study  of  toothed  gear- 
ing, on  the  contrary,  believing  their  investigations  and 
calculations  to  be  the  foundation  upon  which  have  been 
built  the  present  more  abstruse  theories,  their  investi- 
gations have  been  omitted,  because  they  may  be  found 
in  almost  any  comprehensive  work  on  the  subject,  and 
because  it  is  thought  unnecessary  to  repeat  them  here. 

§  II.  —  Proper  Form  of  Tooth- Profiles. 

Let  C  and  Cr  (Fig.  8)  be  two  circles,  in  contact  at 
the  point  a.  If  the  circle  C  be  made  to  revolve  in  the 
direction  indicated  by  the  arrow,  the  circle  Cf  will  be 
made  to  revolve  in  an  opposite  direction  by  the  friction 


10 


TOOTHED   GEARING. 


Fig. 


between  the  two  circles,  supposing,  of  course,  the  fric 
tion  to  be  great  enough  to  overcome  the  resistance. 
Suppose,  now,  it  is  required  of  the  circle  C'  to  perform 
work,  for  example,  to  lift  a  weight 
W  by  means  of  a  string  wound 
around  its  axle.  By  varying  the 
pressure  of  the  circle  C  upon  C'  at 
the  point  a  of  contact,  the  friction 
between  the  circles  may  be  made 
just  sufficient  for  the  lifting  of  the 
weight  :  the  friction  between  the 
circles  will  then  be  the  smallest 
possible  for  the  given  amount  of 
work.  Also,  if  the  circle  C  is 
driven  by  a  constant  and  uniform 
force,  since  the  resistance  and 
motion  are  constant  and'  uniform, 
the  weight  W  will  be  lifted  by  a 
constant  and  uniform  force,  or,  in 
other  words,  power  and  motion 
will  be  uniformly  transmitted. 
We  may  therefore  conclude,  that  in  order  that  toothed 
wheels  may  work  together  most  uniformly,  with  the 
least  friction  and  wasted  power,  and  with  the  greatest 
durability,  the  tooth-profiles  must  be  such  that  the 
driving  wheel  shall  cause  the  driven  wheel  to  revolve 
as  if  moved  by  simple  contact.  If  the  circle  a  roll,  in 
the  direction  indicated  by  the  arrow,  upon  the  circum- 
ference of  the  circle  B  (Fig.  9),  the  point  o  of  the  circle 
a  will  assume  successively  the  positions  o't  o",  o'",  etc., 
the  arc  p'o  being  equal  to  the  arc  p'o',  the  arc  p"o  being 
equal  to  the  arc  p"o",  etc.,  and  the  position  o  of  the 


TOOTHED    GEARING. 


T  I 


point  o  corresponding  to  the  position  b  of  the  rolling 
circle,  etc.  The  point  of  contact,  o,  generates  during 
the  rolling  the  curve  o  o'o"o'"o"" ,  obtained  by  drawing  a 


curve  through  the  successive  positions  of  the  point  o. 
This  curve,  described  by  a  point  of  the  circumference 
of  circle  which  rolls  upon  the  circumference  of  another 
circle,  is  called  an  epicycloid.  In  the  same  manner,  if 


a  circle  a  roll,  in  the  direction  of  the  arrow,  within  the 
circumference  of  another  circle  B  (Fig.  10),  the  point  o 
on  the  circumference  of  the  rolling  circle  will  generate 
;  the  arcs  p'o',  p"o",  and  pmo'"  being 


the  curve  o  o'o"o'" 


12  TOOTHED   GEARING. 

respectively  equal  to  the  arcs  p'o,  p"o,  and  /"'<?,  and  the 
positions  0',  o",  and  o'"  of  the  point  o  corresponding  to 
the  positions  b,  c,  and  d  of  the  rolling  circle  a.  This 
curve,  described  by  a  point  on  the  circumference  of  a 
circle  which  rolls  ivithin  the  circumference  of  another 
circle,  is  called  a  hypocycloid.  In  Fig.  9  the  motion 
of  the  circle  a,  relative  to  the  motion  of  the  circle  B,  is 
precisely  similar  to  the  motion  of  the  circle  C,  relative 
to  the  motion  of  the  circle  Cr  (Fig.  8).  For  in  Fig.  9 
equal  arcs  of  the  rolling  circle  are  developed,  in  equal 
times,  upon  the  circumference  of  the  circle  B ;  and  the 
same  is  true  of  the  circles  C  and  C'  (Fig.  8).  Conse- 
quently the  motion  of  any  point,  as  o,  of  the  circle  a, 
with  reference  to  the  motion  of  the  corresponding  point 
o  of  the  circle  B  (Fig.  9),  must  be  similar  to  the  motion 
of  the  point  a  of  the  circle  C  with  reference  to  the 
motion  of  the  corresponding  point  a  of  the  circle  C' 
(Fig.  8).  But  we  have  shown  that  the  point  o  of  the 
circle  a  (Fig.  9)  generates  an  epicycloid  with  reference 
to  the  motion  of  the  point  o  of  the 
circle  B :  hence,  also,  in  Fig.  8,  the 
point  a  of  C  generates  an  epicycloid 
with  reference  to  the  motion  of  the 
point  a  of  C '.  For  the  same  reasons, 
the  point  a  of  the  circle  O,  revolving 
about  its  fixed  centre,  and  thereby 
causing  the  circle  Of  to  revolve  about 
its  centre  (Fig.  1 1),  generates,  with  reference  to  the 
motion  of  the  point  a  of  the  circle  Of,  a  hypocycloid. 

Let  C  and  C  (Fig.  12)  be  two  teeth,  contact  between 
which  has  just  begun,  C  being  the  driving,  and  C'  the 
driven  tooth.  It  is  plain,  from  what  has  been  said, 


TOOTHED   GEARING.  13 

that  the  motion  of  the  point  a  of  the  tooth  C',  with 
reference  to  the  motion  of  the  point  a  of  the  tooth 
C,  is  similar  to  the  path  described  by  the  point  a  of 
a  circle  O',  which  rolls  within  the  circumference  of  the 
circle  O.  This  path,  as  before  explained,  is  a  hypocy- 
cloid ;  and  consequently,  if  we  give  to  the  portion  ab 
of  the  tooth  C  (called  the  flank  of  the  tooth)  a  hypocy- 
cloidal  form,  the  profile  ab'  will  slide  along  it  with  the 
least  possible  friction.  While  the  point  a  of  the  tooth 
Cr  slides  along  the  profile  ab,  the  point  a  of  the  tooth  C 


also  slides  along  the  profile  ab',  and  generates,  with  re- 
spect to  the  motion  of  the  point  a  of  the  tooth  C',  the 
epicycloid  ab',  the  path  described  by  the  point  a  of 
a  circle  (9",  which  rolls  upon  the  circumference  of  the 
circle  O.  If,  therefore,  we  give  to  the  portion  ab'  of 
the  tooth  C  (called  the  face  of  the  tooth)  an  epicycloidal 
form,  the  profile  ab  will  slide  along  it  with  the  least 
possible  friction.  Again  :  let  the  teeth  of  the  wheels  O 
and  (7  be  in  contact  at  the  point  /  (Fig.  13),  and  suppose 
(J  to  be  the  driver.  The  driving-force  of  the  wheel  C? 
will  be  transmitted  to  the  wheel  O  through  the  point  /, 


TOOTHED   GEARING. 


and  in  the  direction  of  AB,  the  common  normal  to  the 
surfaces  in  contact  at  the  point  /.  From  the  centres 
O'  and  O  draw  the  lines  O'A  and  OB,  each  perpen- 
dicular to  AB.  Let  F  denote  the  driving-force  of  the 
wheel  O',  or  the  force  exerted  by  the  circumference  Y, 
and  F'  the  force  exerted  by  the  point  A.  From  the 
principles  of  the  simple  lever,  we  have  the  propor- 


tion  F\F'  '.'. 


Hence   F'  = 


Since 


or 


Ff  = 


O'A 

the  lines  O'A  and  OB  are 
parallel,  the  perpendicular 
AB  will  be  tangent  to  two 
circles  drawn  with  0  and 
O  as  centres  and  O'A  and 

PB  as  radii,  and  the  force 
'  of  the  point  A  will  be 
directly  transmitted -to  the 
point  B  through  the  line 
AB.  Let  P  denote  the 
force  transmitted  to  the 
circumference  X.  As  be- 
fore, we  shall  have  the 
proportion  P\F' \\OB\Oa, 

From     this    we    obtain 


F:P::OaXOfA:O'aXOB.  From  the  right  -  angled 
triangles  cOB  and  cO'A  we  may  write  the  proportion 
A  (7:  OB\:  cO1  \cO,  which,  multiplied  by  aOiaC?:: 
aO  :  aa,  gives  OA  X  Oa  :  OBX  O'a  : :  cO'X  Oa  :  cOX  Va, 
and  consequently  we  shall  have  F\P :  \cCfxOa : 
cOxO'a.  But,  for  best  results,  all  the  force  of  the 
wheel  Cf  must  be  transmitted  to  the  wheel  O ;  also,  in 


TOOTHED   GEARING. 


Fig. 14 


order  that  the  wheels  may  move  as  simple  friction 
wheels,  the  velocities  at  the  circumferences  must  be 
equal.  Hence  the  forces  F and  P  must  be  equal;  and 
we  will  consequently  have  cO'^aO  —  cOXaO',  which 
can  only  be  true  when  the  points  c  and  a  coincide,  and 
form  one  point.  We  may  conclude  from  this,  that  the 
most  advantageous  form  for  the  profiles  of  the  teeth  is 
such  that  the  common  normal  to  the  profiles  at  the 
point  of  contact  will  pass  through  the  point  of  inter- 
section of  the  line  of  centres  with  the  pitch  circles  X 
and  F.  This  point  is 
called  the  pitch  point. 
Suppose,  now  (Fig.  14), 
the  pitch  circle  O  and 
the  rolling  or  generating 
circle  O'  to  be  regular 
polygons,  having  each 
an  infinite  number  of 
sides.  As  the  polygon 
O'  rolls  in  the  direction 
shown  by  the  arrow,  the 
point  A  generates  an 
epicycloid ;  and  there  is,  for  an  instant,  a  rotation  of  the 
polygon  O'  about  the  point  C.  The  point  A,  for  that 
instant,  describes  an  arc  of  a  circle,  the  centre  of  which 
is  the  point  C,  and  the  radius  of  which  is  the  line  CA. 
But,  since  the  radius  of  a  circle  is  always  normal  to  the 
circumference  at  the  point  of  their  intersection,  the  line 
CA  is  a  normal  to  the  epicycloid  at  the  point  A :  it  also 
passes  through  the  pitch  point  C.  These  two  demon- 
strations were,  we  believe,  first  given  by  M.  Camus  in 
his  "  Cours  de  Mathematiques."  By  a  similar  course  oi 


l6  WOTHED   GEARING. 

reasoning  it  may  be  proved  that  the  normal  CA'  of  the 
hypocycloid  BA'  (generated  by  the  point  A'  of  the  poly- 
gon O",  which  rolls  within  the  polygon  O),  at  the  point 
A'  passes  through  the  pitch  point  C.  If,  now  (Fig.  13), 
we  give  to  the  face  of  the  tooth  of  O'  an  epicycloids! 
form,  and  to  the  flank  of  the  tooth  of  0  a  hypocycloichJ 
form,  the  point  of  contact  of  the  teeth  will  be  the  point 
of  contact  of  two  infinitely  small  circle-arcs,  the  radii 
of  which  are  parallel,  coincide  to  form  the  common 
normal,  and  pass  through  the  pitch  point  a.  We  may 
now  briefly  sum  up  our  arguments  in  order,  and  the 
conclusions  which  must  be  drawn  from  them.  We  have 
shown  (Fig.  8),  that,  in  order  that  the  teeth  of  wheels 
work  most  uniformly  together  and  with  the  least  detri- 
mental friction  possible,  the  action  of  the  driving  wheel 
upon  the  driven  wheel  must  be  such  that  the  wheels 
shall  move  as  if  driven  by  simple  contact.  We  have 
also  proved  (Fig.  12)  that  this  desired  action  takes 
place  between  the  teeth  when  the  faces  of  the  teeth 
are  given  the  epicycloidal  and  the  flanks  of  the  teeth  the 
hypocycloidal  form.  Further:  we  have  proved  (Fig. 
13)  that  the  condition  necessary  for  uniform  power  and 
velocity  is  that  the  common  normal  to  the  teeth  in  con- 
tact, at  the  point  of  contact,  shall  pass  through  the 
pitch  point,  and  (Fig.  14)  that  this  condition  is  fulfilled 
by  teeth  having  epicycloidal  faces  and  hypocycloidal 
flanks.  From  these  demonstrations  but  one  logical  con- 
clusion can  be  drawn, — that  teeth  having  epicycloidal 
faces  and  hypocycloidal  flanks  fulfil  all  the  conditions 
required  of  gear-teeth,  and  that  the  desired  form  of 
tooth-profile  has  been  determined.  Roomer,  the  cele- 
brated Danish  astronomer  and  inventor,  is  said  to  have 


TOOTHED   GEARING.  I/ 

been  the  first  to  demonstrate  the  advantages  of  these 
curves  for  tooth-profiles.  But  De  la  Hire, — who  is 
credited  with  having  first  discovered,  that,  "  if  the  pro- 
files of  the  teeth  of  one  wheel  have  an  epicycloidal 
form,  the  profiles  of  the  teeth  of  its  fellow  will  prop- 
erly have  the  form  of  a  hypocycloid  the  generating 
circle  of  which  has  the  same  diameter  as  that  of  the 
epicycloid  forming  the  teeth  of  the  first  wheel,"  *  - 
Brewster,  Young,  Buchanan,  and  Reuleaux  have  been 
the  chief  promoters  of  the  application. 

Our  investigation  has  now  given  us  the  required 
forms  of  tooth-profile  ;  but  since  these  curves,  like  all 
others,  are  susceptible  of  a  considerable  number  of 
variations,  it  remains  to  determine  somewhat  more 
specifically  the  conditions  upon  which  their  applica- 
bility to  wheel-teeth  depends.  In  the  first  place,  then, 
the  amount  of  curvature,  or  amount  of  deviation,  of 
epicycloidal  and  hypocycloidal  curves  from  the  diameter 
of  the  primitive  or  pitch  circle,  which  passes  through 
the  pitch  point,  depends  upon  the  diameter  of  the  gen- 
erating circle  and  upon  the  diameter  of  the  primitive 
circle,  or,  in  other  words,  upon  the  ratio  of  the  diame- 
ter of  the  generating  circle  to  that  of  the  primitive 
circle.  Thus,  in  Fig.  15,  the  epicycloids  <?,  b,  and  c 
were  generated  by  circles  having  diameters  respectively 
equal  to  \,  f ,  and  \  the  diameter  of  the  primitive  circle 
OO' ;  and  the  hypocycloids  af,  b',  and  c'  had  for  generat- 
ing circles  respectively  the  same  as  the  epicycloids. 
If  d  denote  the  diameter  of  the  generating  circle,  and 
D  that  of  the  primitive  circle,  it  is  plain  from  the 

*  Mr.  J.  I.  Hawkins's  translation  of  Camus  on  the  Teeth  of  Wheels. 


i8 


TOOTHED   GEARING. 


figure,  that,  as  the  ratio  jz  becomes  smaller,  the 

tion  of  the  curve  from  the  diametral  line  Bcf,  passing 
through  the  pitch  point/,  becomes  greater.  If  the  diam- 
eter of  the  generating  circle  of  a  hypocycloid  is  equal 
to  one-half  the  diameter  of  the  primitive  circle,  the 
curve  described  will  be  a  straight  line  coinciding  with 
the  diameter  of  the  primitive  circle  passing  through 

Fig.  15 


a'.- 


the  starting  position  of  the  generating  point.  To 
prove  this,  let  C'  (Fig.  16)  be  the  generating  circle,  and 
C  the  primitive  circle.  Let  o  be  the  starting  position 
of  the  generating  point.  Since  the  diameter  of  Cf  is 
equal  to  one-half  that  of  C,  the  circumference  of  C' 
will  be  equal  to  one-half  the  circumference  of  C,  one- 


TOOTHED    GEARING. 


half  circumference  of  Cr  —  one-fourth  circumference 
of  C,  one-fourth  circumference  of  Cr  =  one-eighth  cir- 
cumference of  C,  etc.  Then,  when  the  circle  Cr  rolls 
sufficiently,  the  point  A  will  fall  upon  the  point  A' ; 
Ao  —  one-fourth  circumference  of  C'  •=.  oAf  •=.  one- 
eighth  circumference  of  C.  The  diameter  AB  will  then 
have  the  position  A'o" ;  the  diameter  oo"  will  have  the 
position  A"o ,  at  right  angles 
to  A'o" ;  and  the  point  o  will 
have  the  position  o'  on  the 
diameter  oE.  Again  :  when 
the  circle  C'  rolls  sufficiently, 
the  point  o"  will  fall  upon  the  d ] 
point  d ;  arc  oAo"  =  one-half 
circumference  of  C'  =  arc 
0./4V  —  one-fourth  circumfer- 
ence of  C.  The  diameter  o"o 
will  then  have  the  position 
do",  and  the  point  o  will  have  the  position  o",  still  on 
the  diameter  oE.  Thus  it  may  be  proved,  that,  for  any 
position  of  the  generating  circle  C',  the  point  o  will 
fall  upon  the  diameter  oE,  and  consequently  that  diam- 
eter is  the  path  of  the  point ;  or  the  curve  generated 
by  the  point  o  will  coincide  with  the  diameter  of  the 
primitive  circle,  which  passes  through  the  starting 
position  of  the  point.  If,  therefore,  we  use  for  the 
generating  circle  of  the  tooth-profiles  one  which  has 
for  a  diameter  one-half  that  of  the  primitive  circle,  the 
flanks  of  the  teeth  will  be  simply  radial  straight  lines, 
as  is  sometimes  the  case  in  practice. 

Fig.  17  shows  forms  of  tooth-profile  for  different  gen- 
erating circles.     Thus  profile   \a\  was  generated  by  a 


2O 


TOOTHED   GEARING. 


circle  having  one-half  the  diameter  of  the  primitive  cir- 
cle O(J\  the  generating  circle  of  profile  20,2  had  for  a 
diameter  three-eighths  that  of  the  primitive  circle ;  and 
the  diameter  of  the  generating  circle  of  profile  3^3  was 
one-quarter  that  of  OOr.  In  profile  3^-3  the  inclination 
of  the  faces  is  so  great,  that  there  may  be,  by  the  princi- 
ples of  the  inclined  plane,  a  tendency  to  produce  press- 
ure upon  the  axles  of  the  wheels  ; 
while  profile  \a\  is  a  weak  form 
for  teeth,  being  narrowest  at  the 
base  \b,  where  it  should  be  widest, 
because  this  part  of  the  tooth 
bears  the  greatest  strain  when  in 
action.  Profile  lai  is  also  a  bad 
form  for  wear,  because  the  friction 
between  the  face  of  one  tooth  of 
this  form  and  the  flank  of  another 
is  much  greater  than  it  would  be  if  the  face  and  flank 
were  more  nearly  envelopes  of  each  other.  Therefore, 
for  greater  strength,  less  friction  and  wear,  and  best 
action  between  the  teeth,  we  should  take  for  the  diam- 
eter of  our  generating  circle  less  than  one-half  and 
greater  than  one-quarter  of  the  diameter  of  the  primi- 
tive circle.  Generating  circles  of  one-third  the  diam- 
eter of  the  primitive  circle  give  very  good  results  in 
practice. 

Let  us  investigate  the  subject  of  profiles  further. 
Let  op p' .  .  .  pvi  be  a  string,  wound  around  the  circum- 
ference of  the  circle  C,  and  fastened  at  the  point  pvi 
(Fig.  1 8).  If,  now,  the  string  be  unwound  from  the 
point  o,  and  held  rigid  as  it  unwinds,  the  end,  or  point 
ot  will  assume  successively  the  positions  </,  o"9  </",  etc.; 


TOOTHED   GEARING. 


21 


the  line  po'  being  equal  to  the  arc  po,  p'o"  —  p'o, 
p"o"r  —  p"o>  etc.  The  curve  o  o'o"o"r .  .  .  ovf,  gener- 
ated by  a  point  of  a  string  as  it  unwinds  from  the  cir- 
cumference of  a  circle,  is  called  an  involute  to  the  circle, 
or  an  involute  simply.  Suppose  (Fig.  19)  the  primi- 


Fig.18 


01V 


tive  circle  to  be  a  regular  polygon,  having  an  infinite 
number  of  sides.  As  the  string  bao'  unwinds,  there 
will  be,  for  an  instant,  a  revolution  about  the  point  a ; 
and  the  point  o'  of  the  string  will  then  generate 
a  circular  arc  having  its  centre  in  the  point  a,  and 

Fig.  19 


a  radius  ao'.  Therefore,  as  was  shown  in  Fig.  14  for 
the  epicycloid,  the  involute  also  fulfils  the  condition 
necessary  for  uniform  power  and  velocity.  For  this 
reason  the  involute  curve  has  been,  and  still  is,  exten- 
sively used  for  tooth-profiles,  the  curve  forming  the 


22 


TOOTHED   GEARING. 


whole  profile,  cd  (Fig.  20) ;  or  the  teeth  having  involute 

faces  and  radial  straight  flanks, 
as  in  ab.  We  have  now  two 
kinds  of  tooth-profile,  cycloidal 
^  and  involute  ;  each  having,  it  is 
presumable,  its  advantages  and 
its  disadvantages  in  practice. 
A  comparison  between  the  two 
is  therefore  necessary. 

§  III.  —  Comparison.  — Advantages  and  Disadvantages  of  Cycloidal  and 
Involute  Teeth. 

Cycloidal  teeth  have  a  great  advantage  over  involute 
teeth,  in  that  the  number  of  teeth,  for  wheels  of  the 
same  diameter  gearing  together,  may  be  reduced  to 
seven,  without  in  any  degree  interfering  with  the 
uniformity  of  action.  Reuleaux  gives  the  smallest 
number  of  involute  teeth  necessary  for  proper  action, 
eleven.  In  cycloidal  teeth  the  loss  of  power  and  wear 
due  to  friction  is  not  so  great  as  in  involute  teeth  ; 
also  the  effect  upon  the  action  of  the  teeth  by  wear 
is  less  in  cycloidal  than  in  involute  teeth,  because  the 
wear  is  evenly  distributed  in  the  former,  and  the  teeth, 
even  when  considerably  worn,  present  more  nearly  the 
original  form  of  profile.  Involute  teeth,  on  the  other 
hand,  have  the  advantage  of  being  easier  and  cheaper 
to  construct  than  the  compound  profiles  of  cycloidal 
teeth.  They  are  also  stronger  for  the  same  width  on 
the  pitch  circle.  Again  :  the  axles  of  wheels  having 
involute  teeth  may  be  moved  slightly  from  or  toward 
each  other  without  disturbing  the  proper  action  ;  while 
a  very  slight  alteration  of  the  distance  between  the 


TOOTHED   GEARING.  2$ 

axles  of  cycloiclal  gears  destroys  the  accuracy  of  motion. 
Straight  flanks  are  acknowledged  by  all  to  be  poor 
forms,  both  on  account  of  their  weakness,  and  loss  of 
work  by  friction.  They  should  never  be  used  except 
for  large  wheels,  where  the  distance  of  the  centres  from 
the  pitch  circles  renders  them  more  nearly  parallel,  and 
consequently  stronger.  The  principal  objection  offered 
to  involute  teeth  is,  that,  especially  in  small  wheels, 
the  great  obliquity  of  the  profiles  tends  to  produce  a 
pressure  upon  the  journals  and  bearings,  as  before 
noticed.  Considerable  difference  of  opinion  exists  as 
to  the  truth  of  this  objection,  and  of  late  years  actual 
experiment  seems  to  assert  its  falsity.  The  following 
experiments  were  tried  by  Mr.  John  I.  Hawkins,  and 
are  taken  from  his  English  translation  of  that  portion 
of  M.  Camus's  "  Cours  de  Mathematiques  "  relating  to 
the  teeth  of  wheels.  Simi-  Fifl,2i 

lar  experiments  tried  by  the  N 
author  of  this  book,  with 
wheels  carefully  sawed  out 
of  black  walnut,  gave  es- 
sentially the  same  results. 
The  approach  noticed  by  Mr.  Hawkins  in  his  Experi- 
ment II.,  however,  failed  to  appear  in  the  experiments 
of  the  author.  Having  constructed  the  sectors  of  two 
wheels,  —  each  of  two  feet  radius,  and  each  containing 
four  teeth  of  the  same  curve  as  those  shown  in  Fig.  21, 
—  one  of  the  sectors  (No.  i)  was  mounted  on  a  fixed 
axis,  and  the  other  on  an  axis  so  delicately  hung,  that  a 
force  of  even  a  few  grains  would  cause  the  axis  of  the 
latter  to  recede  from  that  of  the  former  in  a  direct  line. 
The  following  experiments  were  then  made:  — 


24  TOOTHED   GEARING. 

EXPERIMENT   I. 

The  teeth  of  both  sectors  being  engaged  their  full 
depth  of  an  inch  and  a  half,  No.  i  was  moved  forwards 
and  backwards  a  great  number  of  times,  without  exhib- 
iting the  least  tendency  to  thrust  No.  2  to  a  greater 
distance,  notwithstanding  the  tangent  to  the  surfaces 
of  the  teeth  in  contact  formed  an  angle  of  nearly  sixteen 
degrees  with  the  line  of  centres.  The  points  of  contact 
of  the  teeth  at  the  line  of  centres  were  three-quarters  of 
an  inch  from  the  ends  of  the  teeth. 

EXPERIMENT   II. 

The  teeth  were  engaged  an  inch  and  a  quarter  deep : 
consequently  the  ends  of  the  teeth  were  a  quarter  of 
an  inch  free  from  the  bottoms  of  the  spaces ;  the 
tangent  of  contact  made  an  angle  of  nearly  seventeen 
degrees  with  the  line  of  centres  ;  and  the  point  of  con- 
tact at  the  line  of  centres  was  five-eighths  of  an  inch 
from  the  ends  of  the  teeth.  The  sector  No.-  i,  being 
repeatedly  moved  forwards  and  backwards,  sometimes 
caused  sector  No.  2  to  approach,  but  never  to  recede. 
In  Experiment  I.  the  approach  could  not  take  place, 
because  the  teeth  were  engaged  their  full  depth. 

EXPERIMENT   III. 

The  teeth  were  engaged  one  inch  deep,  leaving  half 
an  inch  between  the  ends  of  the  teeth  and  the  bottoms 
of  the  spaces.  The  angle  of  the  tangent  of  contact 
with  the  line  of  centres  was  eighteen  degrees ;  the 
points  of  contact  at  the  line  of  centres  were  half  an 
inch  from  the  ends  of  the  teeth.  On  the  sector  No.  i 
being  moved  frequently  forwards  and  backwards,  no 
motion  of  the  axle  of  No.  2  appeared, 


TOOTHED   GEARING.  2$ 

EXPERIMENT    IV. 

The  teeth  of  the  sectors  were  engaged  three-quarters 
of  an  inch  deep  :  consequently  the  ends  of  the  teeth 
were  three-quarters  of  an  inch  free  from  the  bottoms 
of  the  spaces ;  the  points  of  contact  of  the  teeth  at 
the  line  of  centres  were  three-eighths  of  an  inch  from  the 
ends  of  the  teeth  ;  the  angle  of  the  tangent  of  contact 
v,-ith  the  line  of  centres  was  nineteen  degrees.  The 
axle  of  sector  No.  2  neither  approached  nor  receded 
on  numerous  trials  made  by  moving  No.  i. 

EXPERIMENT   V. 

The  teeth  were  engaged  half  an  inch  deep ;  the  point 
of  contact  was  a  quarter  of  an  inch  from  the  ends  of 
the  teeth  at  the  line  of  centres  ;  the  ends  of  the  teeth 
were  one  inch  from  the  bottoms  of  the  spaces  ;  the 
tangent  of  contact  formed  an  angle  of  full  twenty 
degrees  with  the  line  of  centres.  In  a  great  number 
of  repetitions  of  this  experiment,  a  slight  receding  of 
sector  No.  2  sometimes  appeared. 

EXPERIMENT   VI. 

The  teeth  were  engaged  a  quarter  of  an  inch  :  the 
ends  of  the  teeth,  therefore,  were  one  inch  and  a 
quarter  from  the  bottoms  of  the  spaces  ;  and  the  points 
of  contact,  one-eighth  of  an  inch  from  the  ends  of  the 
teeth  at  the  line  of  centres ;  the  angle  of  the  tangent 
of  contact  with  the  line  of  centres  was  rather  more 
.than  twenty-one  degrees.  In  this  experiment,  which 
was  repeated  very  frequently,  a  tendency  to  recede 
appeared  several  times,  but  so  slightly  as  to  be  of  no 
practical  importance.  The  quiescent  state  of  the  axle 
was  much  oftener  manifest  than  the  receding. 


26  TOOTHED   GEARING. 

" These  experiments,"  says  Mr.  Hawkins,  "tried 
with  the  most  scrupulous  attention  to  every  circum- 
stance that  might  affect  the  results,  elicit  this  important 
fact,  that  the. teeth  of  wheels  in  which  the  tangent  of 
the  surfaces  in  contact  makes  a  less  angle  than  twenty 
degrees  with  the  line  of  centres,  possess  no  tendency 
to  cause  a  separation  of  their  axes  :  consequently  there 
can  be  no  strain  thrown  upon  the  bearings  by  such  an 
obliquity  of  the  tooth."  Such  an  obliquity  as  twenty 
degrees  must,  unless  counteracted  by  an  opposite  force, 
tend  to  separate  the  axes  ;  and,  as  suggested  by  Mr. 
Hawkins,  this  opposite  force  is  most  probably  the  fric- 
tion between  the  teeth,  which  tends  to  drag  the  axes 
together  with  as  much  force  as  that  tending  to  separate 
them.  Of  course  the  friction  between  teeth  sawed  out 
of  wood  is  greater  than  in  metal  teeth  ;  but  Mr.  Haw- 
kins cites  experiments  tried  by  a  Mr.  Clement,  with 
metal  wheels  lying  loosely  upon  a  work-bench,  in  which 
no  tendency  to  separate  the  axes  of  the  wheels  could  be 
noticed.  This  very  serious  objection  to  involute  teeth 
having  once  been  fairly  removed,  then  the  relative 
value  of  the  two  kinds  of  profile  must  depend  upon  the 
action  between  the  teeth  in  each  case,  the  amount  of 
friction  and  wasted  power,  and  the  relative  expense  and 
difficulty  of  construction.  The  fact,  that,  in  cycioidal 
teeth,  less  power  is  lost  in  overcoming  friction  than  in 
involute  teeth,  seems  to  be  well  established,  in  theory 
at  least,  if,  perhaps,  not  so  well  in  practice  ;  but  whether 
or  not  the  gain  in  this  respect  is  sufficient  to  compen- 
sate for  the  additional  expense  of  construction  over  the 
involute  system,  is  still  a  question  which  must  be  finally 
settled  by  practice  and  actual  experiment.  In  this 


TOOTHED   GEARING.  27 

practical  age,  the  value  of  any  one  mechanism,  com- 
pared with  that  of  another,  is  simply  a  comparison 
between  the  relative  amounts  of  work  to  be  obtained 
from  them  and  the  relative  costs  ;  and  that  system  of 
tooth-profiles  from  which  can  be  obtained  "the  most 
work  for  the  least  money "  must  eventually  gain  the 
supremacy.  In  Fig.  18,  while  generating  the  involute 
curve,  as  fast  as  any  portion  of  the  string  is  unwound, 
it  is  held  rigid,  and  forms  a  straight  line  tangent  to 
the  circle  at  the  point  of  contact ;  as,  for  instance,  the 
portion  pivov  is  tangent  to  the  circle  C  at  the  point  piv. 
Since  this  portion  is  that  which  generates  the  curve, 
and  upon  which  alone  a 
the  curve  depends,  we 
may  assume  the  whole 
string  to  be  rigid  and 
straight,  and  the  re- 
sult will  be  the  same,  a' 
Let  oa  (Fig.  22)  be  a  a. 
straight  line,  which 
rolls  from  right  to 
left  upon  the  circumference  of  the  circle  C.  When 
the  line  oa  has  rolled  sufficiently,  the  point  b  will  fall 
upon  the  point  /  (the  arc  op  being  equal  to  the  line  ob), 
and  the  point  o  will  take  the  position  o' .  When  the 
line  has  rolled  sufficiently,  the  point  b'  will  fall  upon 
the  point  /'  (the  arc  p'o  being  equal  to  the  line  b'o\ 
and  the  point  o  will  then  take  the  position  o".  When 
the  point  b"  falls  upon  the  point  /",  the  point  o  will 
take  the  position  o"\  etc.,  and  the  curve  o  o'o"o'"  thus 
generated  will  be  an  involute  to  the  circle  C.  Thus  we 
have  generated  an  involute  by  rolling  a  straight  line 


28  7VOTHED   GEARING. 

upon  the  circumference  of  a  circle.  But  a  straight  line 
is  the  circumference  of  a  circle  the  radius  of  which  is 
infinitely  long;  and  the  curve  generated  by  a  point  of 
a  circle  which  rolls  upon  the  circumference  of  another 
circle  is  an  epicycloid  :  consequently  an  involute  curve 
is  simply  an  epicycloid  the  generating  circle  of  which 
has  an  infinitely  long  radius  ;  or,  in  other  words,  the 
involute  is  but  a  limiting  case  of  the  epicycloid.  Thus, 
without  coming  to  any  actual  decision  as  to  the  relative 
mechanical  value  of  these  two  curves,  or  rather  two 
different  forms  of  the  same  curve,  we  have,  neverthe- 
less, the  satisfaction  of  having  verified  our  former  con- 
clusion, and  may  still  assert  that  the  cycloidal  form  of 
tooth-profile  fulfils  all  the  conditions  and  requirements, 
and  is  therefore  the  most  useful  and  advantageous. 

§  IV.  —  Practical  Methods  for  laying  out  Teeth,  Exact  and 
Approximate. 

Because  of  the  difficulty  with  which  exact  epicycloidal 
and  hypocycloidal  profiles  are  constructed,  approximate 
methods  are  very  generally  used  ;  and  they  are  found  to 
answer  the  practical  purpose  very  well.  Any  one  of 
the  following  approximate  methods  will  give  very  good 
results,  and  will,  in  ordinary  cases,  answer  as  well  as  the 
more  difficult  and  tedious  exact  methods,  also  given 
here  for  use  in  special  cases  :  — 

METHOD  i  (exact).  —  Let  O  (Fig.  23)  be  the  primitive 
or  pitch  circle.  Take  the  diameters  of  the  rolling  cir- 
cles C  and  K,  each  equal  to  one-third  the  diameter  of  the 
pitch  circle.  Strike  the  circles  C',  C" ,  C'" ,  etc.,  which 
represent  the  different  positions  of  the  rolling  circle  (7, 
and  from  the  points  of  tangency,  b,  b',  b">  etc.,  measure 


TOOTHED   GEARING. 


29 


off  the  following  arcs  :  bp'  —  bp,  b'p"  =  b'p,  b"pr"  —  b"p, 
etc.  The  points  /',/",/"',  etc.,  thus  found,  are  points 
of  the  epicycloid  which  is  to  form  the  face  of  the  tooth  ; 
and  the  curve  pprp"  .  .  .  p*v,  drawn  through  them,  is 
the  face-profile.  For  the  hypocycloidal  flank,  after 
having  struck  the  circles  £/,  Of> ',  O'" y  lay  off  the  arcs 
da'  =  dp,  d'a"  =  d'p,  etc.,  from  the  points  of  tangency 
d,  d',  etc.  The  curve  pa' a"  a'",  drawn  through  the 
points  a't  a",  a'",  thus  found,  is  the  hypocycloid  which 

Fig. 23 


is  to  form  the  flank  of  the  tooth.  The  other  profile, 
xyzt  which  is  similar  to  the  one  just  found,  is  found 
by  starting  at  the  point  y  (py  being  the  given  width 
of  the  tooth  at  the  pitch  circle),  and  rolling  the  gen- 
erating circles  in  the  opposite  directions  from  those 
just  described.  We  have  now  but  to  limit  the  tooth 
at  the  top  and  bottom  by  circle-arcs,  AA  and  BB,  and 
the  profile  is  complete. 

METHOD  2  (exact}.  — In  Fig.  24  O  is  the  pitch  circle, 


3O  TOOTHED   GEARING. 

Cf  and  O"  the  rolling  circles,  and  A  the  pitch  point. 
Divide  the  pitch  circle  and  rolling  circles  into  an  equal 
number  of  small  parts,  equal  each  to  each,  as  shown  in 
the  figure.  Let  the  point  5  of  O'  correspond  to  point  5' 
of  O,  the  point  e  of  O"  correspond  to  the  point  e  of  O, 
etc.  From  A  and  5'  as  centres,  with  5  5'  and  the  chord 
A$  respectively  as  radii,  describe  arcs  intersecting  in 
the  point  c ;  then  from  the  centres  A  and  4',  with  the 
radii  4  4'  and  ^4,  describe  arcs  intersecting  at  <:',  etc. 

Fig.24 


The  points  thus  found  are  points  of  the  epicycloid  Ac'c. 
Similarly,  for  the  hypocycloid,  from  the  points  A  and 
/,  with  radii  e'e  and  Ae,  describe  arcs  intersecting  at 
the  point  /,  and  thus  determine  the  curve  Ap'p.  In 
these  two  methods,  the  closer  together  the  positions 
of  the  rolling  circles  and  the  points  of  division  of  the 
pitch  and  rolling  circles  are  taken,  the  more  accurate 
will  be  the  curves.  When  either  of  these  methods  is 
used,  the  work  of  laying  out  the  teeth  may  be  greatly 
simplified  by  accurately  working  out  one  entire  profile 


TOOTHED   GEARING.  31 

upon  a  smooth  piece  of  wood,  and  cutting  out  this 
profile  for  a  template  with  which  to  trace  the  profiles 
around  the  pitch  circle. 

METHOD  3  (approximate}.  —  From  the  points  i',  2',  3', 
etc.,  a',  bf,  cf,  etc.  (Fig.  24),  as  centres,  and  with  the 
corresponding  chords  of  the  rolling  circles  as  radii, 
draw  circle-arcs.  Thus  the  radius  for  centre  5'  is  A$t 
for  centre  3'  is  A 3,  for  centre  ef  is  Ae,  etc.  The  en- 
velope of  these  arcs,  or  the  curve  which  is  tangent  to 
them,  is  very  nearly  the  correct  profile  of  the  tooth. 

Fig.25 


METHOD  4  (approximate).  — In  Fig.  25  let  A  A  be  the 
pitch  circle,  and  B  and  C  the  rolling  circles.  Let,  also,  / 
be  the  pitch  point,  and  te  and  tk  the  heights  of  the  tooth 
above  and  below  the  pitch  circle.  Take  /;/  =  fte,  and 
strike  through  ;/  the  arc  ;/;/  concentric  with  the  pitch 
circle.  Step  off  on  the  pitch  circle  to  —  tu,  and  from  o 
as  a  centre,  with  the  chord  n't  for  a  radius,  strike  an  arc 
cutting  ;/;/  in  the  point  /.  Draw  po.  The  point  /  is  a 
point  of  the  epicycloid,  and  po  is  the  normal  to  the 
curve  at  the  point  p.  Find  now,  upon  the  line  pol 


32  TOOTHED   GEARING. 

the  centre  for  an  arc  passing  through  the  points  /  and 
/.  In  the  figure,  o'  is  this  centre,  and  o'p  is  the  radius 
for  the  faces  of  the  teeth.  The  centres  for  all  the  faces 
are  upon  the  circle  aa  drawn  through  o't  and  concentric 
with  the  pitch  circle.  Similarly,  for  the  flanks,  take 
tm  —  |  tk,  strike  the  arc  mm' ,  step  off  tb  =  tin',  and, 
with  the  centre  b  and  radius  bt,  strike  an  arc  cutting 
mm'  in  the  point  x.  Draw  xb,  and  find  the  centre  b' 
for  an  arc  passing  through  /  and  x.  The  radius  for  the 
flanks  is  b'x ;  and  the  centres  are  all  upon  the  circle  ddy 
drawn  through  tf,  and  concentric  with  the  pitch  circle. 
METHOD  5  (approximate}.  —  Let  A  (Fig.  26)  be  the 

pitch  circle,  and  a  the  pitch 
point.  Draw  af  tangent  to 
the  pitch  circle  at  the  pitch 
point,  and  make  it  equal  to 
0.57  the  diameter  of  the  roll- 
ing circle,  or  \\  times  the 
circular  pitch  of  the '  teeth. 
Draw  dfe  parallel  to  the 
diameter  aO,  make  df  =  af, 
and  ef  —  the  diameter  of  the 
rolling  circle.  Draw  Od  and 
O  Oep,  and,  taking  ab  =  ac  — 

\af,  draw  bp  and  gpr  parallel  each  to  af.  The  point  /  of 
the  intersection  of  Op  and  bp  is  the  centre  for  the  flank 
ax.  Make  p'c  =  eg,  and  /'  is  the  centre  for  the  face  ay. 
As  before,  all  the  face  centres  are  upon  a  circle  drawn 
through  /'  concentric  with  the  pitch  circle,  and  ali  the 
flank  centres  are  upon  a  circle  drawn  through  p. 

METHOD  6  (approximate}.  —  Let  A  (Fig.    27)   be  the 
pitch  circle,  C  and  B  the  rolling  circles,  and  a  the  pitch 


TOOTHED   GEARING. 


33 


point.  Draw  a'c'b  and  cB'd  through  the  centres  of  the 
rolling  circles,  each  making  angles  of  30°  with  the  line 
of  centres.  Draw  the  line 
cabf  through  the  points  c 
and  b,  and  join  a'  and  d 
with  the  centre  O.  The 
points  g  and  f  are  the 
centres  for  the  face  bx 
and  flank  cy  respectively. 
These  approximate  meth- 
ods are  from  Reuleaux's  A' 
"  Const ructeur,"  and  Un- 
win's  "  Elements  of  Ma- 
chine Design,"  and  are  as 
accurate  as  any  in  use  at 
the  present  time.  When 
a  set  of  wheels  is  to  be 
constructed  so  that  any 
wheel  of  the  set  will  gear 
with  any  other,  the  same 
generating  circles  must  be 
taken  for  all  the  teeth  of 
the  set.  Sometimes  the 
generating  circle  is  taken 
with  a  diameter  equal  to 
the  radius  of  the  smallest 
wheel  of  the  set.  The 
following  are  some  of  the 
simpler  and  rougher  meth- 
ods of  approximation  in 
use :  they  are  convenient  and  easy,  but  give  poor  re- 
sults, and  should  only  be  used  in  rough  work.  Fig.  28, 


34 


TO  OT PI  ED   GEARING. 


draw  ac,  making  75°  with  the  line  of  centres  bd,  and 
make  be  equal  to  one-tenth  the  pitch  of  the  teeth  multi- 
plied by  the  cube  root  of  the  number  of  teeth.  Take 
ab  =  ^bc :  c  is  the  centre  for  the  face  bi,  and  a  the  centre 
for  the  flank  bk.  The  following  values  of  ba  and  be  give 

better    results:    ba  —  -        - — -,  and  be  =  o. 1 2p\lNt    in 

2N  —  20 

which  /  represents  the  pitch,  and  N  the  number  of 
teeth.  In  Fig.  29,  oo  is  the  pitch  circle,  and  bb  and  aa 
the  circles  which  limit  the  teeth  at  top  and  bottom. 

Fig. 29 


The  centres  for  both  faces  and  flanks  are  taken  upon 
the  pitch  circle ;  the  flank  centre  for  gk  and  mn  being 
in  the  centre  of  the  tooth  width  at  x,  and  the  face 
centre  for  cd  and  ef  being*  in  the  centre  of  the  space 
width  at  y.  Still  another  rough  rule  is  to  take  the 
centres  upon  the  pitch  circle,  and  take  the  radius  for 
the  faces  equal  to  one  and  one-fourth  times  the  pitch, 
making  the  flanks  radial  straight  lines. 


TOOTHED   GEARING. 


35 


For  laying  out  involute  teeth,  the  exact  method  is  as 
follows :  Fig.  30,  O  is  the  circle  of  the  bottoms  of 
the  teeth,  and  /  the  starting-point  of  the  involute,  or  the 
root  of  the  tooth.  Lay  off  the  distances  pp ',  p'p",  p"pf", 
etc.,  along  the  circle  OO ;  draw  the  tangents  /Y,  p"a", 
etc. ;  and  step  off  p'a  —  arc  //,  p"ct'  =  arc  />,  p'"a'"  = 
p'"p,  etc.  The  curve  a  a" a'",  etc.,  drawn  through  the 
points  thus  found,  is  the  true  involute  profile.  In 
the  same  manner,  the  profile  cf  is  found,  and  the  tooth 
limited  in  height  by  the  circle  bb.  Radial  straight 
flanks  are  often  used  in  involute  teeth ;  but,  for  reasons 
already  given,  they  should  never  be  used  except  for 
large  wheels,  and  even  then  only  for  rough  work. 
True  involute  profiles 
may  be  easily  traced  by 
means  of  a  straight 
spring  arranged  to  hold 
a  pencil,  or  other  mark- 
er, at  one  end,  and  fas- 
tened  at  the  other  end 
to  the  circumference  of 
a  wooden  circle-segment  of  the  same  radius  as  the  bot- 
tom or  root  circle  of  the  teeth  of  the  wheel.  Because  of 
the  comparative  ease  with  which  true  involute  profiles 
may  be  traced,  approximate  or  circle  arc  methods  are 
not  much  in  use.  The  following  methods,  however, 
give  very  close  approximations  to  the  true  curve,  and 
are,  perhaps,  more  in  use  than  any  others.  In  Fig.  31  ei 
is  the  working  height  of  the  tooth,  i.e.,  the  actual  height 
less  the  clearance  between  the  end  of  the  tooth  of  one 
wheel  and  the  bottom  of  the  corresponding  space  of 
the  other  wheel,  and  im  is  the  actual  height.  Make 


TOOTHED   GEARING. 


ea  =  \ei,  and  draw  ad  tangent  to  the  circle  A  ;  make 
pd  —  \ad,  and  /  is  the  centre  for  the  profile  bak.  A 
circle  through  /,  concentric  with  the  circle  A,  gives 
the  positions  of  the  centres  for  all  the  profiles.  The 

part  kc  may  be  a  straight  line 
tangent  to  bak  at  k,  since  the 
profile  which  engages  with 
bak  does  not  touch  this  part 
at  all.  It  is  better,  however, 
to  round  this  part,  as  in  kf, 
for  greater  strength  and  bet- 
ter casting.  Let  O  (Fig.  32) 
be  the  pitch  circle,  c  and  d 
the  circles  limiting  the  tooth 
at  top  and  bottom  (top  circle 
and  root  circle],  and  a 
the  pitch  point.  Draw 
the  straight  line  ap 
through  the  pitch  point, 
and  making  angles  of 
75°  with  the  line  of  cen- 
V0  tres  ;  draw  fp  through 
the  centre  /,  and  per- 
pendicular to  ap;  and  p 
is  the  centre  for  the 
profile  shown  in  the  fig- 
ure. For  small  teeth, 
the  centres  are  often 
taken  on  the  pitch  circle,  and  the  radius  taken  equal 
to  the  pitch  of  the  teeth. 


Fig.32 


TOOTHED   GEARING. 


§  V.  —  Rack.  —  Internal  Gears. 


37 


If,  in  a  pair  of  gear-wheels,  we  assume  the  radius  of 
one  of  the  pitch  circles  to  be  infinitely  long,  this  pitch 
circle  becomes  a  straight  line  tangent  to  the  other 
pitch  circle  at  the  pitch  point,  and  the  wheel  becomes 
a  rack.  The  rolling  circles  which  generate  the  tooth- 
profiles  for  the  rack  now  roll  along  a  straight  line  in- 
stead of  upon  and  within  the  circumference  of  a  circle, 
and  consequently  the  faces  and  flanks  of  the  teeth 
are  no  longer  epicycloids  and  hypocycloids,  but  both  are 
ordinary  cycloids.  Fig.  33  represents  one  of  the  exact 
methods  for  tracing  the  teeth.  OO  is  the  pitch  circle, 

Fig. 33 


and  a  the  pitch  point.  The  generating  circles  roll  in 
the  directions  indicated  by  the  arrows,  and  the  points 
a,  a",  br,  etc.,  are  found  as  in  Fig.  23  ;  the  arc/V  being 
equal  to  p'a,  p"a"  —  p"a,  rb"  —  ra,  etc.  The  approxi- 
mate methods  for  cycloidal  teeth,  explained  in  the  pre- 
ceding paragraph,  are  applicable  to  the  rack,  Some  of 
these  we  give  as  examples, 


TOOTHED   GEARING. 


Fig.  34 


METHOD  4  (approximate).  —  Let  A  (Fig.  34)  be  the 
pitch  circle,  B  and  C  the  rolling  circles,  and  /  the  pitch 
point.  Let  also  //  and  tk  be  the  heights  of  the  tooth 

above  and  below  the 
pitch  circle.  Take  /;/ 
=  f//,  and  draw  ;/;/,  cut- 
ting the  rolling  circle  C 
in  the  point ;/'.  Step  off 
to  —  arc  /;/,  and  from 
;^7  o  as  a  centre,  with  the 
chord  in'  as  a  radius, 


strike  an  arc  cutting  ;/;/ 
in  the  point  /.  Draw 
po,  and  on  it  find  the 
centre  o'  for  an  arc  of  a 
circle  passing  through  the  points  /  and  /.  It  is  obvious 
that  the  curvature  of  the  flank  will  be  the  same  as  that 

of  the  face.  Therefore,  to 
find  the  flank  centre,  make 
o"o"'  =  omo',  draw  o"b  par- 
allel to  the  pitch  circle  A, 
and  make  ab  =  xo':  b  is 
— A  the  flank  centre.  The 
centres  for  all  the  flanks 
will  be  on  the  line  o"b, 
and  all  the  face  centres  will 
be  on  the  line  </;;/,  drawn 
through  the  points  o'  and 
bt  and  parallel  to  the  pitch 
circle  A. 

METHOD  5  (approximate). — Fig.  53,  A  is  the  pitch  cir- 
cle, and  a  the  pitch  point.    Take  af=o.$?  the  diameter 


V 


TOO  THED   GEA  RING. 


Fig.36 


of  the  rolling  circle,  and  through  /  draw  dfe  parallel  to 

the  line  of  centres.     Take  ab  —  ac  •=.  \af,  and  draw  bd 

and  p'g  parallel  to  AA  :  d  is  the  flank  centre.     Make 

cp'  =  eg,  and/'  is  the  face  centre.     Method  6  of  the  pre- 

ceding paragraph  is  greatly 

simplified   when  applied  to 

the  rack.    The  lines  Odfand 

Oga'  (Fig.  27)  become  paral- 

lel to  the   line  of   centres, 

c'd  and  a'p'  (Fig.   36),   and 

intersect  the  line  cf  in  the 

points  d  and  /,  where  this 

line    meets    the    3O-degree 

lines,*    giving  these  points 

as  centres  for  the  profiles. 

Hence   this    method,   when 

used  for  rack  teeth,  reduces 

to  the  following  :  AA  is  the  pitch  circle,  a  the  pitch 

point,  and  o  and   o'  the   rolling  circles.     Through   the 

centres   o  and   o    draw   the    lines   a'od  and  cfo'p',  each 

making  angles  of  30°  with  the  line  of  centres.     The 

points  d  and  p'  ',  in  which  these 

lines  meet  the  circumferences 

of  the  rolling  circles,  are  the 

centres    respectively   for    the 

flank  p'x,  and  face  dx*. 

When    involute    teeth    are 
used   for  a  rack,  the  profiles 


FFg.37 


\_7 


reduce  to    straight   lines,   making   angles  of   75 

the  pitch  circle  (Fig.  37).     This   may  be  very  prettily 


*  This  is  true  only  when  the  rolling  circles  are  equal. 


40  TOOTHED   GEARING. 

demonstrated  by  means  of  the  approximate  method  of 
Fig.  32  in  the  preceding  paragraph,  as  follows  :  Let  OO 
(Fig.  38)  be  the  pitch  circle,  a  the  pitch  point,  and  ap 
the  75-degree  line.  Since  the  centre  of  the  pitch  circle 
is  infinitely  distant  from  the  pitch  point,  the  perpendic- 
ular pfy  which  passes  through  this  centre,  will  also  be 
infinitely  distant  from  the  pitch  point.  The  radius  ap 
of  the  profile  will  therefore  be  infinitely  great,  and  the 
profile  a  straight  line  perpendicular  to  this  radius,  and 
passing  through  the  pitch  point.  But  since  the  line  ap 

makes  angles  of  75°  with 
the  line  of  centres,  the  per- 
pendicular ab  will  make 
angles  of  75°  with  the 
pitch  circle,  which  is  per- 
"°  pendicular  to  the  line  of 
centres. 

In  internal  gears,  the 
curves  forming  the  faces 
and  flanks  of  the  teeth  are 
reversed  as  compared  with 
external  gears  ;  that  is,  the 
faces  are  hypocycloids,  and  the  flanks  epicycloids.  The 
exact  method  for  constructing  internal  cycloidal  teeth 
is  shown  in  Fig.  39.  O  is  the  pitch  circle,  a  the  pitch 
point,  c'  and  c  the  rolling  circles,  and  o"  and  o'  the  top 
and  root  circles.  Find  the  profile  bad  by  rolling  the 
circles,  as  in  Fig.  23  ;  find,  in  similar  manner,  the  profile 
gxf  (ax  being  the  given  width  of  the  tooth  at  the  pitch 
circle),  and  the  tooth  b'ad'fxg  is  complete.  The  ap- 
proximate methods  given  for  external  cycloidal  teeth 
are  applicable,  without  change  or  difference,  to  internal 


TOOTHED    GEARING. 


gears,  remembering,  however,  that  the  faces  are  hypo- 
cycloidal,    and    the    flanks    epicycloidal    curves.       The 

Fig. 39 


Fig.40 


following,  for  example,  is  method  5,  of  the  preceding 
paragraph  applied  to  internal  gear-teeth.  The  centres 
p  and  /'  (Fig.  40),  for  the 
profiles  ax  and  ay  respec- 
tively, are  found  as  before 
explained  (see  Fig.  26), 
the  curves  drawn,  and  the  o 
tooth  limited  by  the  top 
and  root  circles  O"  and 

a. 

In  generating  involute 
teeth  for  internal  gears, 
the  primitive  circle,  upon 
which  the  generating  line 
rolls,  or  from  which  the  string  unwinds,  may  be  taken 
the  same  as  the  top  circle  of  the  teeth  with  very  good 


42  TOOTHED  .GEARING. 

results.  Thus  in  Fig.  41,  for  the  exact  method,  A  is 
the  pitch  circle,  T  and  R  the  top  and  root  circles. 
Find  the  profile  ca,  as  before  explained.  (See  the 
preceding  section,  Fig.  30.)  In  a  similar  manner  find 

the  profile  de,  and  the  tooth 
is  complete.  The  approxi- 
mate methods  for  external 
involute  teeth  may  be  used 
without  change  for  internal 
gears.  Internal  gears  were 
formerly  quite  extensively 
used ;  but  of  late  years  they 
have  come  to  be  considered 
as  clumsy  contrivances,  and 
are  rarely  used  except  in  special  mechanisms. 

§  VI.  —  Special  Forms.  —  Lantern-Gears.  —  Mixed  Gears. 

This  paragraph  has  been  translated  from  the  French 
edition  of  Professor  Reuleaux's  valuable  work,  "  Le 
Constructeur."  Straight  lines  are  often  used  for  the 
profiles  of  the  teeth  of  gear-wheels,  the  straight  line 
forming  the  flank  of  the  tooth,  and  a  curve  the  face. 
But  teeth  obtained  thus  do  not  gear  together  with  the 
necessary  exactness,  and  for  this  reason  ought  not  to 
be  used  in  the  construction  of  ordinary  machinery.  In 
the  teeth  of  clock-work  gears,  this  kind  of  profile  can 
be  advantageously  used  ;  because  it  permits,  at  the  same 
time,  of  the  easy  cutting-out  of  the  spaces  with  a  file, 
and  of  the  use  of  a  small  number  of  teeth.  If  we  take 
the  diameter  of  the  generating  circle  greater  than  a 
certain  fraction  of  the  radius  of  the  corresponding  prim- 
itive circle,  we  obtain  teeth  which  are  still  of  a  possible 


TO  O  THE  r>   GEA  RJiVG. 


43 


execution,  but  which,  in  practice,  are  admissible  only 
for  particular  cases.  If  we  ^ake,  for  the  generating 
circle,  the  pitch  circle  of  one  ot  the  wheels,  we  obtain, 
for  the  profiles  of  the  teeth  of  the  wheel  corresponding 
to  the  pitch  circle  upon  which  it  rolls,  epicycloidal  arcs, 
while  for  the  other  wheel  the  profiles  are  reduced  to 
points.  It  is  in  this  kind  of  profile  that  we  include 
Ian  tern-gears. 

External  Lantern  Gears  (Fig.  42).  —  From  the  pitch 

FIg.42 


point  a  describe  a  circle  having  a  radius  equal  to  \^  the 
pitch.  This  gives  the  profile  of  the  rung,  or  spindle, 
corresponding  to  the  point  a.  The  face  of  the  tooth 
of  the  wheel  R'  is  formed  by  a  curve  parallel  to  or 
equidistant  from  the  epicycloidal  arc  ab,  generated  by 
the  point  a  in  the  rolling  of  the  circle  R  upon  R'  (the 
arc  tb  —  the  arc  to).  The  envelope  of  circles  described 


44 


TOOTHED  GEARING. 


from  different  points  of  ah.  with  a  radius  equal  to  that 
of  the  rung,  gives  the  face  profile  cd:  the  flank  di  is  a 
circle  quadrant.  The  arc  of  contact  coincides  with  the 
circled;  its  length  al,  of  which  the  limit  /  is  deter- 
mined by  the  top  circle  k,  ought  to  be  greater  than  the 
pitch,  and  hence  at  least  i.i  times  the  pitch.  This  last 
value  serves  to  determine  the  height  g  and  the  real 
height  gf  of  the  face. 

Internal    Lantern-Gears    (Fig.    43).  —  The    following 
manner  of   proceeding    is  similar  to  the  one  just  de- 

Fig.43 


scribed  :  The  portion  cd  of  the  tooth-profile  is  found  by 
a  curve  parallel  to  the  hypocycloidal  arc  ab,  generated 
by  the  point  a  in  the  rolling  of  the  circle  R  within  the 
circle  Rf  (the  arc  tb  —  the  arc  to).  The  arc  of  contact 
al  ought  to  be  taken  at  least  equal  to  i.i  times  the 
pitch.  The  flank  ci  is  a  radial  straight  line  connected 
with  the  rim  of  the  wheel  by  a  small  circle  arc. 

In  Fig.  44  the  hollow  wheel  is  the  lantern  :  the  face 


TOOTHED   GEARING. 


45 


cd  is  parallel  to  the  pericycloidal  arc  ab,  generated  by 
the  point  a  in  the  rolling  of  the  circle  Rf  upon  R  (the 

Fig. 44 


Plfl,45 


arc  tb  =  the  arc  to).  The  arc  of  contact  al  ought  to 
be  at  least  i.i  times  the  pitch  :  the  flank  ci  is  a  radial 
straight  line  connected 
with  the  rim  by  a  small 
circle  arc. 

Fig.  45  represents  a 
particular  case  of  Fig. 
43.  We  have  R  =  %R', 
and  consequently  the 
number  of  teeth  in  R 
=  J  the  number  of  teeth 
in  R'  (N  =  ±Nf).  In 
this  case,  N  =  2,  and  N' 
—  4.  The  profile  cd  is 
parallel  to  the  straight 
line  aiy  to  which  the  hy- 
pocycloid  reduces  (the 
arc  ab  =  the  arc  bi) :  al 
is  the  arc  of  contact.  This  arc  is  here  necessarily  greater 
than  the  pitch  :  since,  however,  the  straight  form  of  the 


46  TOOTHED   GEARING 

flanks  of  the  teeth  of  the  wheel  R'  permits  the  sup- 
pression of  all  play  between  the  teeth,  so  that  the  same 
rung  gears  at  the  same  time  with  two  opposite  flanks, 
the  arc  of  contact  may  be  considered  equal  to  twice 
al.  Many  writers  regard  this  kind  of  gear  as  a  special 
mechanism,  since  in  actual  practice  the  rungs  are  mov- 
able rollers  provided  with  axles.  If  in  Fig.  43  we 
consider  the  radius  R'  as  infinitely  long,  we  obtain  the 
mechanism  of  the  rack,  in  which  the  profiles  of  the 
teeth  upon  the  rack  itself  afe  formed  by  curves  parallel 
to  ordinary  cycloids.  If,  again,  in  Fig.  44,  we  consider 
the  radius  R'  as  infinitely  long,  we  obtain  a  very  simple 
form  of  rack,  which  is  very  often  used  in  preference  to 
the  preceding.  Upon  the  pinion  the  profiles  of  the 
teeth  are  formed  by  curves  parallel  to  an  involute  to 
the  pitch  circle.  Lantern-gears,  in  cases  which  require 
a  certain  precision  and  not  very  frequent  use,  offer  the 
advantage  that  the  rungs  can  be  easily  and  exactly 
described  with  a  pair  of  compasses.  Lantern-racks  of 
wrought  iron  are  very  useful  in  practice  for  apparatus 
exposed  to  cold  and  wet ;  such  as  for  lifting  gates,  draw- 
bridges, etc. 

Gear  at  Two  Points  (Fig.  46).  —  If  we  connect  togeth- 
er two  gears  at  a  single  point,  we  obtain  a  new  style 
of  gear,  which  allows  us  to  adopt  for  one  of  the  wheels 
a  very  small  number  of  teeth,  and  consequently  a  great 
difference  in  the  revolutions  of  the  two  wheels,  even 
though  both  wheels  are  quite  small.  In  the  figure  the 
two  pitch  circles  are  at  the  same  time  the  generating 
circles  of  the  profiles  of  the  teeth  :  ac  is  an  epicycloidal 
curve  (generated  by  the  rolling  of  R'  upon  R),  which, 
for  the  length  of  contact  al,  gears  with  the  point  a  of 


TOOTHED   GEARING. 


47 


the  wheel  R' ;  ab  is  a  second  epicycloidal  curve  (gen- 
erated  by  the  rolling  of  R  upon  Rf),  which,  for  the 
length  of  contact  all,  gears  with  the  point  a  of  the 
wheel  R  ;  ai  and  ai'  are  the  profiles  for  the  flanks  of 
the  teeth  for  the  wheels  R'  and  R.  The  small  wheel 
is  used  frequently  for  shrouded  wheels.  This  kind 

Fig.46 


of  gear  is  frequently  met  with  in  cranes  and  hoisting- 
machines. 

Mixed  Gear  (Fig.  47).  — -  This  kind  of  gear,  which 
is  very  convenient  for  the  small  pinions  of  hoisting- 
machines,  has  the  advantage  of  diminishing  the  space 


48 


TOOTHED   GEARING. 


at  the  root  of  the  tooth.  This  result  is  due  to  the  use 
of  radial  straight  lines  for  the  flanks  of  the  teeth  of  the 
small  wheel.  In  order  to  obtain  a  sufficient  duration  of 
engagement,  it  is  convenient  to  use  upon  both  wheels 
the  curves  which  form  the  faces  of  the  teeth  as  far  as 
their  points  of  intersection.  In  the  figure,  ac  is  an  arc 
of  a  cycloid,  or  involute,  generated  by  the  rolling  of  R/ 

Fig.47 


(which  here,  for  a  rack,  is  a  straight  line)  upon  R:  ai'  is 
a  radial  straight  line  generated  by  the  rolling  of  the 
circle  J^upon  the  inside  of  R  (the  radius  of  W=  |  that 
of  R).  The  gearing  of  the  profile  ac  with  the  point  a 
takes  place  for  the  length  of  contact  all.  The  cycloidal 
arc  ab,  generated  by  the  rolling  of  W  upon  R',  gears 
with  the  flank  ai'  for  the  length  of  contact  ai. 


TOOTHED   GEARING. 


49 


Fi?.48 


§  VII.  —  Bevel  Gears. 

The  different  gears  hitherto  described  are  intended 
to  transmit  power  from  one  shaft  to  another  parallel 
shaft.  If  we  wish  to  transmit  from  one  shaft  to  another 
which  is  not  parallel,  or  which  makes  an  oblique  angle 
with  the  first,  we  must  make  use  of  either  bevel  or 
screw  gears.  A  bevel  or 
conical  gear  differs  from 
a  cylindrical  or  spur  gear 
in  that  its  two  pitch  cir- 
cles (at  the  two  ends  of 
the  teeth)  are  of  differ- 
ent diameters,  and  conse- 
quently the  ends  of  any 
one  tooth  are  of  differ- 
ent heights,  widths,  etc. 
The  pitch  circles  of  a 
pair  of  bevel  wheels  limit 
frusta  of  cones,  the  api- 
ces of  which  meet  at  the 
point  of  intersection  of 
the  axes  of  the  wheels. 
Thus,  in  Fig.  48,  o  is  the 
point  of  intersection  of 
the  axes  ox  and  oy  ;  a'b', 
ab>  a'cf,  and  ac  are  the 
pitch  circles;  a'c'ca  and  a'b'ba,  the  "pitch  frusta;"  and 
a' do  and  a'b'o,  the  "pitch  cones."  The  axes  may  make 
any  angle  with  each  other.  It  should,  however,  be 
remarked  that  wheels  such  as  are  represented  in  (c) 
are  seldom  used  in  practice,  since  the  same  angle  may 
be  obtained  with  the  wheels  shown  in  (b).  To  lay  out 


\ 


5<D  TOOTHED    GEARING. 

the  pitch  cones  and  frusta,  we  have,  then,  the  following 
simple  rule :  Draw  a'c  and  a'b'  (Fig.  48),  making  with 
each  other  the  required  angle,  and  equal  respectively  to 
the  diameters  of  the  larger  pitch  circles  of  the  wheels. 
Draw  now  the  axes  oy  and  ox  perpendicular  to  their 
respective  pitch  circle  planes.  The  point  of  intersec- 
tion o  determines  the  cones ;  and  the  given  length  a' a 
or  tfb,  the  frusta.  The  ends  of  bevel  teeth  lie  upon  the 

Fig.49 


surfaces  of  cones  which  are  supplementary  to  the  pitch 
cones,  and  of  which  the  top  and  root  circles  limit  frusta. 
In  Fig.  49,  a"cc'a'"  and  V'dd'b'"  represent  two  teeth  ;  ab 
and  a'b'  are  the  pitch  circles  ;  a'b'o,  the  pitch  cone ;  and 
a'b'ba,  the  pitch  frustum.  The  root  circles  are  a"bn 
and  a'"b"'\  and  the  top  circles,  cd  and  c'd'.  These 
circles  limit  the  frusta  cdb"a"  and  ddrb'"a'"  of  the  sup- 
plementary cones  cdo"  and  c'd'd.  The  larger  of  the 


TOOTHED    GEARING. 


two  pitch  circles  determines  the  size  of  the  wheel  and 
the  tooth  dimensions.     For  instance,  a  fifteen-inch  bevel 


of  one  inch  pitch  means  a  bevel  of  which  the  larger 
pitch  circle  is  fifteen  inches  in  diameter,  and  of  which 


52  TOOTHED    GEARING. 

the  pitch  of   the  teeth  upon  this  pitch  circle  is  one 
inch. 

The  teeth  of  bevels  may  be  either  cycloidal  or  invo- 
lute. The  latter  are,  however,  more  often  used,  because 
they  are  much  easier  to  construct.  Fig.  50  shows  a 
convenient  method  for  laying  out  the  teeth  of  bevels  : 
abo  is  the  pitch  cone ;  pn  and  gi  are  the  top  circles ; 
and  ef  and  kl,  the  root  circles.  Produce  pc  as  far  as 
its  intersection  with  the  axis  of  the  wheel,  and  from 
this  point  o'  as  a  centre,  with  o'a,  o'p,  and  o'e  as  radii, 
describe  the  circle-arcs  q,  /,  and  ;;/.  These  are  the 
virtual  pitch,  top,  and  root  circles.  Find  by  any  of 
the  preceding  methods  the  centres  for  the  faces  and 
flanks,  regarding  a  as  the  pitch  point :  r  and  s  are  the 
centre  circles  for  the  faces  and  flanks  respectively.  The 
teeth  xyy"x"  and  it/t"wt  are  correct  in  size,  and  are 
drawn  to  give  the  pattern-maker  his  dimensions.  Now 
project  and  describe  the  actual  pitch,  top,  and  root 
circles  q',  /',  and  ;//',  also  the  same  circles  for  the  small 
end  of  the  tooth  (q"t  /'' ',  and  ;;/'),  and  the  centre  circles 
r',  /,  etc.  Set  off  now  x'x'"  =  xx",  y'n  =  ak ',  and  z'z" 
=  yy",  and  find  upon  the  centre  circles  the  centres  for 
arcs  passing  through  the  points  xf  and  y'  (face),  and  y' 
and  z'  (flank).  The  widths  of  the  small  end  of  the  tooth 
at  the  pitch,  top,  and  root  circles,  are  determined  by  the 
lines  o"y't  o"x' ',  and  0'V,  etc.,  and  the  faces  and  flanks 
drawn  as  above.  Draughtsmen  sometimes  find  the 
centres  for  the  tooth-profiles  upon  the  actual  instead 
of  upon  the  virtual  pitch  circle,  as  we  have  done ;  but 
the  height  of  the  teeth  upon  the  actual  pitch  circle  is 
less  than  the  real  height,  as  a  glance  at  Fig.  50  will 
show ;  and  consequently  the  widths  upon  the  top  and 


TOOTHED   GEARING. 


53 


root  circles  are  respectively  too  great  and  too  small, 
thus  marring  the  correctness  of  the  drawing.  In  Fig. 
48  (a),  the  planes  of  the  pitch  circles  of  the  two  bevels 
are  at  right  angles  with  each  other  (the  angle  0  =  90°). 
If,  now,  we  gradually  increase  this  angle  6,  the  wheels 
take  the  form  of  Fig.  48  (b)  ;  and  finally,  when  the 
planes  become  parallel  (0  —  180°),  become  external 
cylindrical  or  spur 
gears.  If,  on  the 
contrary,  we  gradu- 
ally decrease  the  an- 
gle 0,  the  bevels  take 
first  the  form  of  Fig. 
48  (c),  and  finally, 
when  the  planes  be- 
come coincident  (6 
=  0),  become  inter- 
nal cylindrical  gears.  Between  these  latter  two  cases 
(Fig.  48  (c)  and  internal  cylindrical  gears)  we  have  two 
interesting,  if  not  altogether  practicable,  cases,  —  the 


Fig.52 


internal  bevel  and  the  disk 
wheel.  A  pair  of  bevels, 
the  internal  bevel  being 
in  section,  is  represented 
in  Fig.  51,  and  the  "disk 
wheel"  in  Fig.  52.  The 
internal  bevel,  because  of 
the  difficulties  in  the  way 
of  its  construction,  is 
never  used  in  practice.  It  may  be  constructed,  however, 
if  desired,  by  the  rules  already  given  for  internal  cylin- 
drical and  bevel  gears.  The  disk  wheel  is  the  least 


54  TOOTHED   GEARING. 

difficult  of  all  bevels  to  construct,  and  is,  although 
seldom  used,  for  this  reason  entitled  to  a  place  among 
practical  gears.  The  disk  wheel  and  pinion  possess 
one  peculiarity  not  found  in  any  other  bevel;  viz.,  the 
ratio  of  the  radius  of  the  pinion  to  that  of  the  disk 
wheel  depends  upon  the  angle  included  between  the 
axes  of  the  wheels.  If  we  let  r  and  R  be  the  radii  of 
the  pinion  and  disk  wheel,  and  tf  the  angle  included 

between  the  axes,  we  shall  have  the  relation  -^  =  cos  tf. 

The  supplementary  cones  upon  which  lie  the  ends  of 
the  teeth  become,  for  the  disk  wheel,  right  cylinders 
having  diameters  equal  to  those  of  the  pitch  circles, 
and,  when  cycloidal  teeth  are  used,  the  profiles  are  or- 
dinary cycloids,  the  disk  wheel  being  regarded  as  (and 
is  sometimes  called)  the  "bevel  rack."  In  order  that 
a  set  of  bevel  gears  shall  gear  together  each  to  each,  it 
is  necessary,  not  only  that  the  pitch  and  kind  of  tooth 
profile  be  the  same,  but  that  the  slant  height  (b'o,  Fig. 
48)  of  the  pitch  cones  be  the  same  in  all  the  wheels  of 
the  set.  Practice,  however,  allows  a  slight  variation 
from  this  rule ;  and,  according  to  Reuleaux,  bevels  will 
work  sufficiently  well  together  if  the  difference  in  the 
lengths  of  these  slant  heights  does  not  exceed  five  per 
cent.  Such  wheels  are  called  bastard  wheels,  and  are 
quite  commonly  used  in  cases  where  there  is  no  neces- 
sity for  very  accurate  gear. 

§  VIII.  —  Screw  Gears.  — Worm  and  Wheel. 

Screw  gears  are  cylindrical  gears,  in  which  the  teeth 
are  not  parallel  to  the  axes  of  the  wheels,  but  make 
oblique  angles  with  them.  All  the  lines  of  the  teeth, 


TOOTHED   GEAKING. 


55 


which  in  spur  gears  arc  parallel  to  the  axes  of  the 
wheels,  are  in  screw  gears  parts  of  helices  drawn 
around  the  pitch,  top,  and  root  circles.  Let  Fig.  53 
represent  two  screw  gears  ;  ®  being  the  angle  included 
between  the  axes,  and  <£  and  </>'  the  angles  made  by  the 
teeth  with  the  "middle  planes  "  of  the  wheels,  as  shown 
in  the  figure.  It  is  plain  that  the  angle  aob  is  equal  to 
the  angle  ® :  conse- 
quently we  have  from 
the  figure,  <£  +  <£'  +  © 
=  1 80°.  This  condi- 
tion must  be  fulfilled, 
else  the  wheels  will 
not  gear  properly  to- 
gether. Another  ne- 
cessary condition  in 
screw  gears  is,  that  ~~ 
the  pitches  of  two 
gears  which  work  to- 
gether, taken  normal 
to  the  directions  of 
the  teeth  or  the  nor- 
mal pitches,  must  be 
equal.  It  is  more 
convenient  to  lay  off  the  pitches  on  the  pitch  circles ; 
that  is,  to  lay  off  the  circumferential  pitches,  instead 
of  the  normal.  In  Fig.  54,  ab  represents  the  normal 
pitch,  and  ae  the  circumferential.  The  angle  aeb  being 

equal  to  <£,  we    have   ae  =  -^ — ,  the    circumferential 

sin  <£ 

pitch  equal  the  normal  pitch  divided  by  sin  <£.     In  order 
that  the  wearing  surfaces  may  be  equal,  the  lengths  of 


50  TOOTHED   GEARING. 

the  teeth  of  a  pair  of  screw  gears  should  be  equal. 
The  width  of  face  depends  upon  the  length  of  tooth 
and  the  angle  <£.  Thus,  in  Fig.  54,  /  =  ec  being  the 
length  of  the  tooth,  and  I'  =  dc  the  width  of  face,  we 
have  the  angle  ced  =:  angle  </>,  and  consequently  /'  —  / 
sin  <£.  Suppose  (Fig.  53),  ®  =  40°,  and  <f>  —  60°  :  hence 
<£'  +  60°  +  40°  =  1 80°,  </>'  =  80°.  If  /  and  /  represent 


Fig.54 


Fig.55 


the  circumferential  pitches,  and  //  the  common  normal 
pitch,  we  shall  have, 


sin 


and 


sin  60°      0.866 


sin  $'      sin  80°      0.985 

Also,  for  the  widths  of  faces  of  the  two  wheels,  we  shall 
have 

/'  =  /  sin  </>  =  0.866/,  and  I"  =  /  sin  <j>'  —  0.9857. 


TOOTHED   GEARING. 


$'/ 


If  we  make  9  =  90°,  we  will  have  an  ordinary  spur- 
wheel   gearing  with   a   screw   gear   (Fig.  55).      In    this 
*igure,    <£  —  90°    and    0  =  40°  :     hence    <f>'  =.  180°  - 
(90°  +  40°)  =  50°.     We  therefore  have,  for  the  circum- 
ferential pitches  and  widths  of  faces, 


=  «,    p  = 


n 
0.766' 


sm  90  sm  50" 

/'  =  /  sin  90°  =  /,    and  /"  =  /  sin  50°   =  0.766^. 
Let  6  —  90°,  that  is,  the 
axes  are   at   right  angles 
with  each  other  (Fig.  56) : 
consequently 

<f>  -f-  9'  =  1 80°  —  90°  =  90°. 

The  angles  9,   9',  may  be 

equal  or  unequal :   in   the 

figure  they  arc  taken  equal. 

9  =  0'  =  9°1  =  45°. 

2 


From  this, 


and  I'  =  I"  =  I  sin  45°  ^  0.7077.*  In  Fig.  57  the  axes 
are  parallel,  or  0  =  o :  hence  <f>  -f-  9'  =  180°.  This  sig- 
nifies that  9  and  9'  are  supplementary,  9  =  180°  —  9'. 
The  inclinations  of  the  teeth  across  the  faces  of  the 
wheels  are  in  opposite  directions.  We  have  taken  9'  = 

60°  :  hence  <A  —  120°,  and  we  have    ^'= =  — -— •• 

-sm6o°      0.866 

*  If  we  make  the  angle  </>  less  than  the  angle  of  repose,  which,  for 
cast-iron  on  cast-iron,  is  about  10°,  only  the  wheel  /"  can  be  the  driver: 
the  wheel  I'  then  restrains  motion  in  the  direction  opposite  to  that  in 
which  it  is  driven. 


TOOTHED   GEARING. 


Since  the  sin.  of  an  angle  equals  the  sin.  of  its  supple 
ment,  /  =  /  and  /'  =  /"  =  /  sin  <£  =  /  sin  $  =  o.86f 

Frg.57 


Screw  Rack  and  Pinion.  —  If  we  make  the  radius  of 
one  of  a  pair  of  screw  gears  infinitely  long  ( —  oo ),  the 

Fig.58  Fig.59 

I  I 


wheel  becomes  a  screw  rack,  and  the  pair  constitutes  a 
screw  rack  and  pinion,  shown  in  Fig.  58.     Let  0  =  45°, 


TOOTHED   GEARING. 


59 


and   </>  = 
;/ 


75°:   hence  $  =  180°  -  (45°  +  75°)  =  60°, 
;/  ;/  ;/ 


_  //___ 

~  sTnT'  "~  5^66'  -  '  sin  * 

=  0.9667,  and  7"  =  /  sin  <£'  =  O.866/. 

Fig.  59  represents  a  spur  rack  gearing  with  a  screw 


pinion,    6  =  45°,    </>  =  90°  :    hence  <//  = 


45°,    /  = 


S1U     tp 

—  7  sin  </>'  =  0.7077. 


—  - 
=  7'    and  l" 


Fig. 6 I 


\ 


\ 


\ 


We  may  also  have  a  screw  rack  gearing  with  a  spur 
pinion,  by  making  </>'  =  90°  (Fig.  60).     Let  0  —  45°,  and 


sin  9 
=  ;/,  I'  —  I  sin  </>  =  0.7077,  and  7"  =  7  sin  <£'  =  7. 

If  we  make  the  radii  of  both  wheels  of  a  pair  of 
screw  gears  equal  to  infinity,  the  pair  becomes  two 
screw  racks  gearing  together  (Fig.  61) ;  and  if  we  make 
</>  or  <£'  =  90°,  we  have  a  spur-rack  gearing  with  a 
screw  rack  (Fig.  62). 


6o 


TOOTH£D   GEARING. 


To  draw  the  tooth  profiles  for  a  screw  gear  we  pro- 
ceed as  follows  :  Having  determined  the  angle  <£  of  the 
teeth,  and  the  length  /,  draw  the  horizontal  line  xy  (Fig. 
63).  Draw  db,  making  the  angle  <£  with  xy,  and  make 
it  equal  in  length  to  /.  Drop  the  lines  dc  and  be  per- 
pendicular respectively  to  xy  and  dc.  The  line  be  is  the 
length  of  the  tooth  projected  in  the  plane  of  the  pitch 
circle  P.  Strike,  now,  the  pitch,  top,  and  root  circles, 
P,  /,  and  r,  and  make  aU  =  be  (a  being  the  pitch  point). 


Fig. 62 


Fig.63 


Find  the  centres  for  faces 
and  flanks,  as  in  spur  gears, 
and  draw  the  profiles  through  a,  b',  and  f.  In  con- 
structing screw  gears,  it  is  advantageous  to  make  the 
angles  of  the  teeth  equal  (<£  =  <//).  The  circumferen- 
tial pitches,  and  tooth  dimensions  in  the  planes  of  the 
pitch  circles,  as  also  the  face  widths,  will  then  be  equal, 
thus  saving  calculation  and  extra  work.  The  friction 
between  the  teeth  is  also  more  evenly  distributed  by 
this  means. 

The  motion  between  two  well-constructed  screw  gears 


TOOTHED   GEARING. 


6l 


Is  very  regular  and  uniform.  They  are  therefore  useful 
in  cases  where  uniformity  of  motion  is  requisite  ;  but, 
owing  to  the  friction  between  the  teeth,  these  gears  are 
not  very  durable,  and  should  be  used  for  the  transmis- 
sion of  small  powers  only,  and  at  comparatively  slow 
motion.  y 

Worm  and  Wheel. — The  mechanism  known  as  the 
worm  and  wheel,  or  the  worm  and  worm-wheel,  is  a 
modification  of  screw 
gears  with  axes  at 
right  angles,  the  prin- 
cipal object  of  which  is 
to  obtain  conveniently 
a  great  difference  in  Fig.  64 
the  revolutions  of  two 
shafts.  The  worm  is  an 
endless  screw,  and  the 
worm-wheel  a  screw 
gear  (Fig.  64).  It  is 
evident  from  the  fig- 
ure, that  (the  worm  being  the  driver),  at  each  revo- 
lution of  the  worm,  the  wheel  will  be  moved  through 
a  distance  equal  to  one  tooth.  Hence,  if  the  wheel 
has  thirty  teeth,  the  worm  will  make  thirty  revo- 
lutions while  the  wheel  makes  one,  or  the  worm-shaft 
will  revolve  thirty  times  as  fast  as  the  wheel-shaft. 
The  common  angle  A  of  the  teeth  is  usually  taken 
such  that  the  worm  will  drive  the  wheel,  while  the 
wheel  will  not  drive  the  worm ;  so  that,  if  at  any 
time  the  driving-power  is  taken  off,  the  gearing  will 
remain  stationary.  For  this  purpose,  the  angle  A.  may 
be  taken  from  4^°  to  9°.  If,  however,  the  worm-wheel 


62  TOOTHED    GEARING. 

is  to  be  the  driver,  X  must  be  taken  greater  than  10°. 
The  pitch  radius  R'  of  the  worm  may  be  from  one  to 
two  times  the  circumferential  pitch.*  The  tooth-pro- 
files of  the  worm  and  wheel  may  be  either  cycloidal  or 
involute  ;  and,  in  either  case,  those  of  the  worm  are 
drawh  as  for  a  rack,  and  those  of  the  wheel  as  for  a 
screw  gear. 

Involute  profiles  are  particularly  useful  in  worms,  be- 
cause the  worm  is,  at  best,  difficult  to  construct,  and  the 
straight  75°  profiles  of  the  involute  rack  very  much  fa^ 
cilitate  the  construction.  If  we  make  the  radius  of  the 
worm-wheel  infinitely  long,  the  wheel  becomes  a  screw 
rack,  and  the  mechanism  becomes  a  worm  and  screw 
rack  (Fig.  65).  We  may  also  have  a  worm  and  internal 
worm-wheel  (Fig.  66),  or  an  internal  worm  and  worm- 
wheel.  In  either  of  these  cases  the  profiles  are  drawn 
as  explained  in  sections  IV.  and  VIII.  As  in  screw 
gears,  by  placing  the  axes  at  oblique  angles,  we  may 
have  a  worm  gearing  with  an  ordinary  spur  wheel,  a 


Fig.,64a  *  If  we  develop  in  the  straight  line  ac  (Fig. 

64  a)  the  circumference  of  the  pitch  circle,  and 
in  the  straight  line  ab  the  length  of  one  revolu- 
tion of  the  screw,  we  shall  have  be  —  the  pitch 
=  /,  and  ac  =  the  circumference  =  2irR'  :  hence 


This  condition  must  be  fulfilled  :  hence,  if  we 

make  R'=  2/,  --f—  \,  tan  A  =  O.I59X  £  =  0.0795, 
A 

A  =  4°  33'.      If  R'  =  p,    ^—  i,    and    tan  1 
—  o.  1  59,   A  =  9°  2'.     Inversely,  if  7i  —  1  2°, 
tan  a  =  0.213  =  0.159-,,   -^=1.34,   R'  =  $p. 


TOOTHED   GEARING. 


Fig. 65 


spur  rack,  or  an  internal  spur  wheel.  It  must,  however, 
not  be  forgotten  that  the  pitch  of  the  spur  gear  must 
be  taken  equal  to  the  pitch  of  the  worm  multiplied  by 
cos  A.  In  order  to  obtain  more  bearing  surface  between 
the  teeth  of  the  worm  and 
those  of  the  wheel,  the  bot- 
toms of  the  spaces  in  the  ^ 
wheel  are  sometimes  cast 
in  the  form  of  circle-arcs,  to 
fit  the  threads  or  teeth  of 
the  worm  (the  radius  of  curvature  equals  radius  of 
ends  of  worm-teeth  plus  the  clearance),  and  the  ends 
of  the  wheel-teeth  formed  to  fit  the  bottoms  of  the 
spaces  in  the  worm  (radius  of  curvature  equals  radius  of 
bottoms  of  worm-spaces  plus  the  clearance),  as  shown 


Fig.66 


Fig.67 


in  Fig.  67.  The  figure  gives  a  section  through  the 
centre  of  the  wheel,  showing  two  teeth  entire  and  an 
end  view  of  the  worm.  As  with  plain  screw  gears,  so 
with  the  worm  and  wheel,  the  wear  is  excessive ;  and, 
for  this  reason,  only  comparatively  small  powers  can  be 
advantageously  transmitted  by  this  mechanism.  In 


64 


TOOTHED   GEARING. 


cases,   however,   where   the   gears  are    not    in    motion 
continuously,    as    in    hoisting-machines,    cranes,    some 


CO 


machine  tools,  etc.,  worms  may  be  used  for  the  trans- 
mission of  considerable  powers. 


TOOTHED   GEARIXG.  65 

§  IX.  — Hyperbolic  Gears.* 

Hyperbolic,  or,  more  properly,  hyperboloidal,  gears 
are  intended  to  be  fixed  upon  arbors,  the  axes  of  which 
cross,  without  intersecting  each  other.  Their  primitive 
surfaces  (surfaces  limited  by  the  primitive  or  pitch  cir- 
cles) are  hyperboloids  of  revolution,  which  touch  along 
a  common  generatrix.  This  generatrix  may  be  deter- 
mined as  follows  :  — 

In  Fig.  68,  which  is  a  projection  made  normally  to 
the  shortest  distance  between  the  axes,  let  us  divide 
the  angle  of  inclination  0  of  the  axes  into  two  other 
angles,  (3  and  ft,  in  such  a  manner  that  the  perpendicu- 
lars AB  and  AC,  drawn  to  some  point  A  of  the  line  of 
division  SA,  shall  be  inversely  proportional  to  the 
numbers  of  revolutions  of  the  wheels,  i.e.,  directly  pro- 
portional to  the  diameters.  SA  is,  then,  the  generatrix 
of  contact  of  the  two  hyperboloids.  AB  =  R'  and 
AC  =  R'  represent  the  projections  of  the  radii  of  two 
normal  sections  through  the  point  A,  and  we  have, 

R'  _sinff  _nf  _  N^ 
57  "  siiiT?  ~~  «  ~  N" 

n  and  ;/  being  the  numbers  of  revolutions,  and  N  and 
N'  the  numbers  of  teeth,  of  the  wheels.  The  real  radii, 
R  and  R,,  are  still  to  be  determined,  as  also  are  the 
radii  SD  —  r  and  SE  =  r'.  Between  these  last  we 
have  the  relation, 

-+COS0 
r  _  tan  p  _  n 

7  =  tan/?""^ 

—f  -h  cos  0 

*  From  Le  Constructeur. 


66  TOOTHED    GEARING. 

That  is,  r  and  r'  are  in  the  same  relation  to  each  other 
as  are  the  two  segments  AF  and  AG,  which  are  deter- 
mined by  the  projections  of  the  axes  upon  the  right 
line  FG,  drawn  through  the  point  A  perpendicular  to 
the  generatrix  of  contact.  Representing  by  a  the 
shortest  distance  between  the  axes,  we  have, 

i-h-cos<9  i+-'cos<9 

r  n  ,  r  n 

and  —  = 


n'  \n 


a  n          „   .    In  V  a  n'        A      /n'\2 

i  -f  2  —  cos  0  -f  (  —, )  i  +  2  —  cos  0  -H  —  J 


n 


The  radii  R  and  Rl  are  the  hypothenuses  of  right- 
angled  triangles,  of  which  the  sides  are  respectively  Rf 
and  xy  =  r,  Rf  and  x'y'  =  r',  and  consequently  have 
the  values, 


R  =  y^2  +  >'2     and     R,  =  \A#/2  +  r'2. 

R'  and  Rt'  are  known  from  what  precedes  when  we 
have  given  the  length  SA  =  !.  The  angles  ft  and  f? 
are  determined  by  the  relation 

-_     and     tan/T= 


n  A  ,. 

—.  +  cos  v  —  -f-  cos  0 

«  « 

As  in  bevel  gearing,  the  problem  permits  of  two 
solutions,  according  as  the  line  SA  is  drawn  withir 
the  angle  0,  or  within  the  supplementary  angle  BSC' 
(Fig.  69).  These  two  solutions  differ  from  each  other 
in  the  direction  of  rotation  of  the  driven  arbor.  One 
of  these  solutions  leads  to  an  internal  gear,  as  in  bevel 
gears ;  but  this,  to  our  knowledge,  has  never  been 
actually  constructed,  and  it  cannot  possibly  have  any 


TOO  THED    GEARING. 


practical  value.     When   the   angle   of  inclination,  0,  is 
made  equal  to  90°,  we  have, 


and 


'-,-»*  B -ffi 

r  \a  I 


a       n2  -f-  ;/2' 


n*  -f  «/ 


It  is  easily  seen,  from  what  precedes,  that  hyperbolic 
gears  present  a  more  limited  number  of  solutions  than 

Fig.69 


ordinary  screw  gears,  with  which,  however,  they  pre- 
sent many  analogies.  In  the  latter,  for  one  value  of 
the  angle  of  inclination  of  the  axes,  we  can  give  an 
arbitrary  value  to  the  angle  of  inclination  of  the  teeth 
of  one  of  the  wheels  ;  while  in  hyperbolic  gears  there  is 
only  one  pair  of  values  admissible  for  the  angles  of 
inclination. 

The  primitive  surfaces  of  two  hyperbolic  gears  are 


68  TOOTHED   GEARING. 

formed  by  corresponding  zones  of  two  hyperboloids  of 
revolution.  When  the  distance  (shortest)  between  the 
axes  is  small,  the  zones  comprising  the  circles  of  the 
gorge,  of  which  r  and  r  (Fig.  68)  are  radii,  cannot  be 
utilized  as  primitive  surfaces,  and  we  must  have  re- 
course to  zones  somewhat  removed  from  these  circles. 
These  may  ordinarily  be  replaced  by  simple  frusta  of 
cones,  and  the  construction  thus  rendered  compara- 
tively simple.  The  following  examples  will  serve  to 
illustrate  the  preceding  formulas  and  remarks  :  — 

Example  I.      0  =  40°,     -  =  -,     a  =  4".     From   the 

R'       ri  Rf       i 

formula  — 7  =  —  we  have  -=-f  =.  -  =  0.5  ; 

RI       n  K.l       2 

also  we  have 

r       0.5  4-  00340°  _  1.266 

7  =:   2  +  cos 40°   :=  ^66  =  °4577 

r  i  4-  2  cos  40°  _  2.532  = 

a       i  -f  2  x  2  cos  40°  -f  4      8.064 

r=  1.2559",     r'  =  2.744". 
For  the  angles  ft  and  fi'  we  have 

o          sin  40°          0.6428 
tan  ft  :=  2  +  cos  40°  =  ^66-  = 

or  ^8—13°  5',  and  jtf  =  40°  —  0  =  26°  55'.  For  the 
distance  SA  =  /  =  8"  we  have 

R'  =  /sin  13°  5'  =  8  X  0.226368  =  1.81" 

^P/  =  8  X  sin  26°  55'  =  8  x  0.452634  =  3.62". 


TOOTHED   GEARING.  69 

Finally, 


R  =Vi^ 
and 

R,  =  N/p^2  +  ^74  2  -  4-54". 

Example  2.      0  —  90°,  —  —  -  (a  value  which  will  be 

satisfied  by  the  numbers  of  teeth  TV^^  36  and  N'  =  20), 
and  #  =  0.8".     From  the  preceding  formulas  we  have 


8  1 


52  4-  92  106 

and 

/  =  o.i  86". 


We  have  also  tan  /?  — —  —  1.80,  or    /8  =  60°  57',    and 

consequently  /^  =  29°  3'.      For  R  =  2"  we   have   the 
formula 


R'  =  \JR2  —  r*  =  \22  -  o.6i2  =  1.90" 

and 


9  9 

Also  for  R,  we  have 


Rt  =  ViTo62  -f-  oT892  =  i.  08". 


70  TOOTHED   GEARING. 


Example^.    0  =  90°,     —  =  i.       As    before,    tan    ft 
=    '=i»   or/2=45°,       =       V=i,   orr  =  /.     Also 


R  •=.  RIt  and  the  hyperboloids  are  congruent. 

Example  4.      In  the  particular  case  where  the  rela- 

tion —  is  numerically  equal  to  cos  0,  and  the  line  of 

division  which  determines  the  angle  ft  is  situated  within 
the  supplementary  angle  of  0  in  such  a  manner,  that,  tak- 

ing into  consideration  the  sign,  we  have  —  =  —  cos  0, 

one  of  the  primitive  surfaces  reduces  to  a  cone,  and  the 
other  to  a  hyperboloidal  plane.  This  hyperbolic  plane 
(or  disk)  wheel  corresponds  to  the  disk  wheel  in  bevel 
gears,  and  can  be  made  to  gear  with  an  ordinary  bevel 
wheel.  It  offers,  however,  no  practical  advantage,  since 
the  disk  wheel  interferes  with  the"  prolongation  of  the 

arbor  of  the  bevel.    For  0  =  60°,    —  —  --  =  —  cos  60°, 

11  2 

T  _ 

we  obtain  the  disk  wheel,  and  have  tan  ft  —  -  y/3,  ft  =  30°, 


R=R',   R,  —  V^/2  +  (i2  —  \/4^2  +  a2.     If  -were  neg- 

ative, and  less  than  cos  0,  we  would  obtain  a  hyperbolic 
internal  gear  ;  but  gears  of  this  kind  are  not  at  all 
practical. 

With  hyperbolic  gears  we  may  obtain,  as  a  limiting 
case,  the  mechanism  of  a  rack  and  pinion.  The  rack, 
m  this  case,  carries  oblique  teeth  ;  while  the  pinion  is 


TOO  THED   GEA RING. 


Fig.70 


formed  by  the  zone  corresponding  to  the  circle  of  the 
gorge  of  a  hyperboloid  of  revolution.  But  since  the 
construction  of  this  pinion  is  much  more  difficult  than 
that  of  a  screw  gear,  the  effect  of  which  is  equivalent, 
it  results  that  the  latter  should  be  used  in  all  cases 
where  this  effect  is  to  be  produced. 

Teeth  of  Hyperbolic  Gears.  —  If  we  wish  to  give  to  the 
teeth  of  hyperbolic  gears  perfectly  accurate  forms,  we 
meet  with  very  serious  difficulties  in  the  execution. 
We  may,  however,  content  ourselves  with  approximate 
forms.  In  this  case,  to  determine  the  teeth  of  a  hyper- 
bolic gear,  we  begin  by  tracing  the  supplementary  cone 
of  the  hyperboloidal  zone,  which 
is  to  be  used  as  the  primitive  sur- 
face. The  apex  H  of  this  cone 
(Fig.  70)  is  obtained  by  drawing 
a  perpendicular  AH  to  the  gen- 
eratrix  SA,  parallel  to  the  plane 
of  the  figure.  We  then  deter- 
mine the  profiles  of  the  teeth 
for  the  normal  pitch  ptl  upon  the 
circle  of  the  gorge  as  if  it  was 
acted  upon  by  a  screw-wheel 
having  a  diameter  r,  and  an  in- 
clination of  teeth  90°  —  /?;  then 
we  continue  the  profiles  thus 
obtained  upon  the  conical  surface  HJL,  taking  care  to 
increase  the  dimensions  parallel  to  the  circle  of  division 
in  the  proportion  of  /  to  pn  (p  being  the  circumfer- 
ential pitch),  and  the  lengths  in  the  proportion  of  K  to 
r,  K  representing  the  length  of  the  generatrix  of  the 
supplementary  cone.  We  repeat  the  same  construction 


72  TOOTHED   GEARING. 

for  the  supplementary  cone  corresponding  to  the  other 
base  of  the  zone,  being  careful  to  decrease  the  values 
of  /  and  K.  Thus  we  obtain  for  each  tooth  two  pro- 
files, sufficiently  exact,  of  which  the  corresponding 
points  must  be  joined  by  straight  lines  to  form  the 
body  of  each  tooth. 

In  certain  cases  a  cone  frustum  may  be  substituted 
for  the  hyperboloidal  zone,  upon  the  condition  of  prop- 
erly determining  the  apex.  To  this  effect,  we  revolve 
the  generatrix  SA  about  the  axis  HS  until  the  point  A 
becomes  coincident  with  the  point  J :  the  projection 
of  the  generatrix,  in  this  position,  determines  by  its 
intersection  with  HS  the  desired  apex  of  the  cone. 

§  X.  —  Relations  between  Diameter,  Circumference,  Pitch,  Number  of 
Teeth,  etc.  —  Diametral  Pitch.  —  Methods  for  stepping  off  the 
Pitch. 

The  circumference  of  a  circle  is  expressed  by  the 
formula 

C=  irD,    or  C=  271-7?         (i) 

where  C  is  the  circumference,  D  the  diameter,  R  the 
radius,  and  TT  the  constant  3.14159.  From  these  formu- 
las we  may  write, 

c  c 

/?  =  -,    R=~          (2). 

TT'  2?r 

Thus,  to  find  the  circumference,  multiply  the  diameter 
by  3.14159,  or  the  radius  by  2  X  3.14159  =  6.28318. 
Inversely,  to  find  the  diameter,  divide  the  circumfer- 
ence by  3.14159:  to  find  the  radius,  divide  the  circum- 
ference by  6.28318.  The  simple,  old  rule,  which  says, 


TOOTHED   GEARING. 


73 


This 


"  To  find  the  circumference  of  a  circle,  multiply  the 
diameter  by  22,  and  divide  by  7,  to  find  the  diameter, 
multiply  the  circumference  by  7,  and  divide  by  22," 
ordinarily  answers  the  purpose  well  enough.  The  cir- 
cumferential pitcJi  or  circular  pitch  (generally  called 
simply  the  pitch)  of  a  gear  of  any  kind  is  the  distance 
from  the  centre  of  one  tooth  to  the  centre  of  an  adja- 
cent tooth,  measured  on  the  pitch  circle,  or,  what  is  the 
same  thing,  the  distance  on  the  pitch  circle,  which 
includes  one  tooth  and  one  space, 
distance,  laid  off  a  certain  num- 
ber of  times  around  the  pitch 
circle,  divides  the  pitch  circle 
into  a  certain  number  of  equal 
parts,  each  containing  one  tooth : 
consequently  the  circumference 
of  the  pitch  circle  divided  by 
the  pitch  will  give  the  number 
of  teeth,  and  the  pitch  multi- 
plied by  the  number  of  teeth 
will  give  the  circumference  of  the  pitch 
formula,  N  being  the  number  of  teeth,  and  /  the  pitch, 


From  formula  (i)  we  may  write  —  —  — ,  and,  from  the 

6       wD 

i      N  i       N 

third  of  formula  (3),  -  =  -^.     Hence  -  =  — — ,  or 


N          7T  7T 

_  =  _=^    and     -  =  /         (4). 


74  TOOTHED    GEARING, 

This  ratio  of  the  constant  quantity  TT  —  3.14159  to  the 
circumferential  pitch  is  called  the  diametral  pitch, 
because  it  is  equal  to  the  ratio  of  the  number  of  teeth 
to  the  diameter  of  the  pitch  circle.  We  represent  this 
diametral  pitch  by  pd.  The  diametral  pitch  gives  the 
number  of  teeth  in  a  gear  wheel  per  unit  (say  inch) 
of  length  of  the  pitch-circle  diameter.  To  illustrate., 
suppose  we  have  a  pitch  circle  of  10"  diameter  and  i 
circumferential  pitch  of  3. 141 59".  From  formula  (i)  the 
circumference  is  C  —  •*  X  10  =  31.41-59",  and  from  for- 
mula (3)  the  number  of  teeth  is  N  =.  —  =±-  ^  =  10. 

P      3.HI59 

Hence,  from   formula   (4),   pd  —  —  =  —  —  i  ;   that   is, 

there  is  one  tooth  in  the  gear  for  each  inch  of  length 
in  the  diameter  of  the  pitch  circle.  In  order  to  distin- 
guish the  diametral  from  the  circumferential  pitch,  the 
former  is  often  designated  as  "pitch  No. — ."  Diame- 
tral pitch  No.  i  =  circumferential  pitch  of  -  =.  3.14159", 

diametral   pitch   No.    2  =  circumferential    pitch   of 

=  i.57079">  etc. 

Since  the  circumference  of  a  circle  cannot  be  meas- 
ured exactly  (the  quantity  TT  being  irrational),  it  is  often 
tedious  work  to  step  off  the  circumferential  pitch  arounc 
the  pitch  circle  (especially  in  large  gears),  a  great  many 
trials  being  necessary  before  the  equal  division  of  the 
pitch  circle  is  obtained.  A  formula,  by  the  use  of  whicl: 
this  work  is  simplified,  may  be  obtained  as  follows  :  Let 
bed  (Fig.  72)  be  a  circle,  be  a  circle  chord.  In  the  tri 
angle  abc  we  have,  from  trigonometry,  the  proportior 


TOOTHED   GEARING. 
sin  angle  bac  \bc\\  sin  angle  bca  :  ab. 


75 


But  ab  =  R,  the  radius  of  the  circle,  and  be  =  I",  the 
circle  chord.  Calling  the  angle  bac  6,  we  have,  since 
ac  =  ab  =  R,  the  relation, 

-  0 

2 


angle  bca  = 


Substituting  these  values  in  the  above  proportion,  we 
obtain 


Hence 


But 


.    /i8o°  - 

sin  I  - 
V        2 


/i8o°-0\            /     0      0\ 
sin  f J  =  sin  I  90 j  =  cos  £  0 


and,  from  trigonometry,  sin  0  =  2  sin  J  6  cos  J  0. 

These  values,  substituted  in  the 
last  expression  for  I'1 ',  give 


Fig.72 


sn 


cos 


cos4<9 


or 


(5). 


Suppose,  now,  the  arc  be  to  repre- 
sent the  pitch  laid  off  on  the  pitch 
circle  of  a  gear.  If  we  represent  by  N  the  number  of 

360° 
teeth  in  the  gear,  we  shall  have  for  6  the  value  6  =         , 


76  TOOTHED   GEARING. 

1  80° 

and  consequently  \  0  =  -^-.     From  this,  by  substitu- 

tion  in  formula   (5),  we   have   for   the   length    of   the 
chord  be, 


(6). 


Rule.  —  To  find  the  length  of  the  chord  subtended 
by  the  pitch  arc,  multiply  the  diameter  of  the  pitch 
circle  by  the  sine  of  the  angle  obtained  by  dividing 
1  80°  by  the  number  of  teeth. 

Example  i.  —  Suppose  D  =  24"  and  N=  80.     Hence 

T  0-.O 

i  o  —  if^L  —  2°  15',-   sin  2°  15'  =  0.03926,  and  /"  =  24 

oO 

X  0.03926  =  0.942". 

Example  2.  D  =  39!"  and  the  pitch  =.  p  •=.  4". 
From  formula  (i)  the  circumference  is  C  =  trD  =  124", 

and    from    formula    (3)  N  =.  —  3—31.      Formula   (6) 

4 

therefore  gives  I"  —  39^  sin  /L8o!\  —  39^  sin  5°  48'  23!" 

V  31  / 

=  392  X  0.1011683  =  3-996//- 

Mr.  W.  C.  Unwin,  in  "  Elements  of  Machine  Design," 
gives  the  following  :  — 

"  To  lay  off  the  Pitch  on  the  Pitch  Line.  —  The  follow- 
ing construction  is  convenient  when  the  wheel  is  so 
large  that  it  is  impossible  to  find  the  exact  pitch  by 
stepping  round  the  pitch  line.  Let  the  circle  (Fig.  73) 
be  the  pitch  line.  At  any  point,  a,  draw  the  tangent  ab. 
Make  ab  equal  to  the  pitch.  Take  ac  equal  to  \ab. 
With  centre  c  and  radius  cb,  draw  the  arc  bd.  Then 
the  arc  ad  is  equal  to  ab,  and  is  the  pitch  laid  off  on  the 


TOOTHED   GEARING.  77 

pitch  line.  When  the  wheel  has  many  teeth,  the  arc 
ad  sensibly  coincides  with  its  chord ;  but,  if  it  has  few 
teeth,  there  is  an  appreciable  error  in  taking  the  chord 
ad  equal  to  the  pitch." 

Unfortunately  neither  of  these  rules  gives  exactly  the 

required  distance;  for,  in  the  first  case,  the  sin  f— — _J 

is  usually  a  number  containing  six,  eight,  or  even  more 
decimal  places,  and  consequently  the  chord  be  will  be 
such  a  number,  not  capable  of  exact  measurement  with 
the  compasses ;  and,  in  the  second  case,  the  pitch 
(being  the  circumference  —  F|  73 

an    irrational     quantity  —  di-        j> c    a 

vided  by  the  number  of  teeth) 
cannot  be  exactly  laid  off  on 
the  line  ab.  Such  simple  and 
easily  remembered  rules,  how- 
ever, simplify  in  some  degree 
the  work  of  the  draughtsman  and  mechanic,  and  are 
therefore  worthy  of  our  notice.  An  accurately  con- 
structed ""Tr-rule"  (pi-rule),  used  in  connection  with  the 
preceding  method,  gives  very  close  results.  To  con- 
struct such  a  rule,  have  a  four-inch  circle  turned,  as 
accurately  as  possible,  out  of  wood  or  metal.  Mark  a 
point  anywhere  upon  the  circumference,  and  starting 
with  this  point  tangent  to  a  straight,  true  ruler  about 
14"  long,  roll  the  circle  along  (taking  care  not  to  slip  or 
slide)  until  the  point  is  again  tangent  to  the  ruler.  The 
distance  thus  developed  upon  the  ruler  is  equal  to  the 
circumference  of  the  4"  circle,  equals  4?r.  Divide 
the  developed  length  into  four  parts  :  each  part  is  equal 
to  one  TT  (pi),  and  may  be  divided  into  halves,  quarters, 


7  8  TOOTHED    GEARING. 

eighths,  etc.,  or  into  tenths  and  hundredths.  The  total 
distance  now  marked  off  is  477,  and  the  divisions  are 
equal  to  TT,  JTT,  JTT,  JTT,  etc.,  or  -^TT,  and  yJ-^TT.  As  an 
example  to  illustrate  the  use  of  the  7r-rule,  suppose 
the  diameter  of  a  gear  to  be  constructed  is  10",  and  the 
number  of  teeth  100.  The  circumference  of  the  pitch 
circle  is  IOTT,  and  the  pitch  is  IOTT  divided  by  100,  or 
-^QTT.  This,  measured  on  the  7r-rule,  and  laid  off  on  the 
tangent  line  ab  (Fig.  73),  will  give  the  arc  ad  (or  chord 
ad)  as  accurately  as  any  method  with  which  we  are 
acquainted. 

§  XI.  —  Ratios.  —  Velocity.  —  Revolution.  —  Power. 
The  velocity  ratio  of  two  gear  wheels  is  the  velocity  at 
the  circumference  of  one  wheel  divided  by  the  velocity 
at  the  circumference  of  the  other,  both  velocities  being 
taken  in  terms  of  the  same  unit  (generally  feet  per 
second),  or  the  ratio  of  the  velocity  at  the  circumference 
of  one  to  the  velocity  at  the  circumference  of  the  other. 
The  velocity  ratio  of  two  toothed  wheels  which  gear 
together  is  always  constant,  and  equal  to  unity ;  that  is, 
the  velocity  at  the  circumference  of  one  is  equal  to  the 
velocity  at  the  circumference  of  the  other.*  To  prove 
this,  let  the  circles  of  Fig.  74  represent  the  pitch  circles 
of  a  pair  of  gear  wheels.  Suppose  R  to  be  the  driver, 

*  When  two  gear  wheels  are  fixed  upon  the  same  shaft,  their  veloci- 
ties are  proportional  to  their  diameters  or  radii.  Thus, 
let  D  and  D'  be  the  diameters  of  two  such  wheels. 
The  velocities  at  the  circumferences  of  the  wheels  are 
v  =  Cn,  and  v'  —  C'n  ;  v,  v',  C,  and  Cf  being  the  velo- 
cities and  circumferences.  Hence 

i/    nzy*    jy    Rr 

-  =  -r_    =:gr  = 


TOOTHED    GEARING. 


79 


Fi8.74 


and  r  the  driven  wheel.  As  the  wheels  revolve,  it  is 
plain,  that,  as  each  tooth  of  R  passes  the  imaginary  line 
AB,  it  carries  with  it  a  tooth  of  the  wheel  r.  Thus 
equal  numbers  of  teeth  of  the  two  wheels  pass  the  line 
AB  in  equal  times.  But,  since  the  pitches  of  the  wheels 
are  equal,  equal  numbers  of  teeth  must  lie  on  equal 
arcs  of  the  two  pitch  circumferences  :  therefore,  with- 
out reference  to  the  relative  sizes  of  the  wheels,  equal 
arcs  of  their  pitch  circumferences 
pass  the  line  AB  in  equal  times,  or, 
in  other  words,  the  velocities  at  the 
circumferences  are  equal. 

The  revolution  ratio  of  two  gear 
wheels  which  gear  together  is  the 
greater  number  of  revolutions  di-  \  \ 
vided  by  the  less,  or  the  ratio  of  the  \ 
greater  number  of  revolutions  to 
the  less.  For  example,  if  one  of  a 
pair  of  gear  wheels  makes  100  revo- 
lutions per  minute  and  the  other  20, 
the  revolution  ratio  is  ^V"  =  i>  anc^ 
we  say  the  wheels  are  geared  5  to  i. 
in  Fig.  74  that  equal  numbers  of  teeth  of  the  wheels  R 
and  r  pass  the  line  AB  in  equal  times.  Let  us  suppose 
the  number  of  teeth  (N)  of  the  wheel  R  to  be  100,  and 
that  (N')  of  r  to  be  25.  When  25  teeth  of-.-/?  have 
passed  the  line  AB,  25  teeth  (all)  of  r  have  also  passed 
the  line ;  that  is,  R  has  made  J  of  a  revolution,  and  r  has 
made  i  entire  revolution.  When  50  teeth  of  R  have 
passed  the  line  AB,  50  teeth  of  r  have  also  passed  the 
line,  or  R  has  made  £  of  a  revolution,  and  r  has  made 
2  entire  revolutions.  Thus,  when  100  teeth  of  R  have 


We  have  proved 


80  TOOTHED   GEARING. 

passed  AB,  or  when  R  has  made  i  entire  revolution,  r 
has  made  *££-  =  4  entire  revolutions.  The  revolution 
ratio  of  the  pair  is  therefore  -*,  the  small  wheel  making 
4  revolutions  while  the  large  wheel  makes  i.  But  the 
ratio  of  the  number  of  teeth  of  the  small  wheel  (r)  to 
that  of  the  large  wheel  (R)  is  -f^  =  ^  :  therefore  it  is 
plain  that  tJie  revolution  ratio  of  a  pair  of  toothed  wheels 
is  inversely  equal  to  the  ratio  of  the  numbers  of  teeth  of 
the  wheels.  Letting  ;/,  N,  R,  D,  and  C  represent  the 
number  of  revolutions,  number  of  teeth,  radius,  diameter, 
and  circumference  respectively,  of  the  smaller  wheel, 
and  ;/,  N',  R',  Df  ,  and  C  the  number  of  revolutions, 
etc.,  of  the  larger  wheel,  we  have,  since  the  number  of 
teeth  is  directly  proportional  to  the  radius,  diameter,  or 
circumference, 

n      N'     R     D      C' 


Rule.  —  The  number  of  revolutions  of  the  smaller 
wheel  is  to  the  number  of  revolutions  of  the  larger 
wheel  as  the  number  of  teeth,  radius,  etc.,  of  the 
larger  wheel  are  to  the  number  of  teeth,  radius,  etc.,  of 
the  smaller  wheel. 

Example  i.  —  Two  bevel  wheels  are  to  gear  together 
so  that  the  revolutions  per  minute  are  respectively 
n  =  1  60  and  ;/  =  40.  The  diameter  of  the  smaller 
wheel  is  D  =  8",  and  the  pitch  of  the  teeth,  p  =  $  '.  It 
is  required  to  find  the  diameter  of  the  larger  wheel  (ZX) 
and  the  numbers  of  teeth  (N  and  N'}  of  each  wheel. 

We  have  here  *  =  —  =  i     From  formula  (7),  i  =  ^ 
;/        40        i  I        8 

D'  =  32".     From  formula  (i),  C  =  *•  X  8  =  25.  i",  and, 


TOOTHED   GEARING.  8 1 

from   formula  (3),  N  =  -~-  —  50.     From  formula  (7), 

2 

again,  ^  —  — ,  or  N'  =  200. 

Example  2.  —  A  shop  shaft  makes  120  revolutions  per 
minute.  From  this  shaft  it  is  required  to  gear  down 
to  8  revolutions  per  minute.  The  diameter  of  the 
wheel  on  the  first  shaft  is  12" .  Find  other  diameters 
and  the  numbers  of  teeth  of  each  wheel,  supposing  the 

Fig. 75 


pitch  =  i".     The  revolution  ratio  is  152=11.     From 

o  I 

formula  (7),  I*  =  — ,     D'  —  1 80"  =  1 5  feet.     A  wheel 

of  this  size  is  out  of  the  question  :  we  therefore  must 
have  recourse  to  a  train  of  wheels  such  as  is  repre- 
sented in  Fig.  75.  We  may  take  the  revolution  ratio 
between  D  and  D'  \ ,  and  that  between  D"  and  D'"  \  : 
we  then  have  f  X  f  =  ^  as  the  ratio  between  D  and 

D'".     From  formula  (7),  then,  -t  =  3  =  — .    ZX  =  36", 

and  n~  =  1  =  ^.  Taking  Z>/x  =  D  =  12",  we  have 
D"'  =  6o".  From  formula  (i),  C=  *  X  12  =  37.7,  and, 


82  TOOTHED   GEARING. 

from  formula  (3),  JV=  ^ZiZ  =38.     Hence,  from  formula 

(7),  N'  =  1 14,     N"  =  38,     and    N"'  =.  190.* 

In  a  pair  of  gears  in  which  N=  25  and  IV' =  100 

the  revolution  ratio  is  — ,  —  —  =  —  —  -.     The  same 

;/      TV        25        i 

teeth  are  therefore  in  contact  once  in  every  revolution 
of  the  larger  wheel,  or  once  in  every  4  revolutions  of 
the  smaller  wheel.  Contact  taking  place  so  frequently 
between  the  same  two  teeth,  if  these  teeth  happen  to 
be  rough  and  poor,  the  wear  between  them  must  be 
greater  than  in  any  other  part  of  the  wheels.  If,  how- 
ever, we  make  N=  26,  the  revolution  ratio  is  Ye0'—  3iJ> 
practically  the  same  as  before,  and  the  two  poor  teeth 
are  in  contact  only  once  in  13  revolutions  of  the  larger, 
or  50  revolutions  of  the  smaller  wheel.  By  means  of 
this  "wear  tooth  "  the  wear  of  the  wheels  may  be  more 
evenly  distributed,  and  the  durability  of  the  wheels  con- 
siderably increased,  without  seriously  interfering  with 
the  revolution  ratio  of  the  wheels. 

Power  Ratio.  —  The  power  or  force  of  a  gear  wheel 
is  the  force  with  which  the  circumference  of  the  wheel 
.turns  :  it  is  equal  to  that  force,  which,  when  applied 
to  the  circumference  in  a  direction  contrary  to  that  of 
rotation,  is  just  sufficient  to  stop  the  rotation  of  the 
wheel.  The  power  ratio  or  force  ratio-  of  two  gears 
is  the  greater  power  divided  by  the  less,  or  the  ratio  of 
the  greater  power  to  the  less.  The  powers  of  two 
wheels  which  gear  together  are  equal,  the  power  of  the 

*  The  gears  D'  and  Z>",  being  fixed  upon  the  same  shaft,  of  course 
make  the  same  number  of  revolutions  per  minute,  regardless  of  diameters 
or  radii. 


TOOTHED   GEARING. 


Fig.76 


driver  being  transmitted  directly  to  the  driven  wheel : 
in  this  case,  therefore,  the  power  ratio,  as  the  velocity 
ratio,  is  constant,  and  equal  to  unity.  Let  R  and  R' 
(Fig.  76)  represent  the  radii  of  a  pair  of  gears,  and  ;' 
the  radius  of  a  pulley  which  is  fixed  upon  the  axle  of  R, 
and  arranged  to  lift  a  weight  W  by  means  of  a  string 
passing  around  its  circumference.  Let  the  power  or 
force  of  the  driver  R'  be  denoted  by  P.  This  force  is 
transmitted  to  R  in  the  direction  shown  by  the  arrow. 
We  may  regard  the  imaginary  line  ac  as  a  simple  lever, 
the  fulcrum  of  which  is  at 
b,  and  the  arms  of  which  are 
ab—r  and  bc=.R,  The 
force  P  acts  upon  the  long 
arm,  and  the  force  W  upon 
the  short  arm.  By  the  prin- 
ciples of  the  lever,  the  mo- 
ments of  the  forces  with 
reference  to  the  fulcrum 
must  be  equal  :  hence  we 
have 

W     R 
Wr  =  PR,     or    -p  =  -          (8). 

That  is,  the  forces  of  the  wheels  R  and  r  are  inversely 
proportional  to  their  radii.  Since  the  radii  R  and  r  are 
directly  proportional  to  the  velocities  of  the  circumfer- 
ences, and  the  power  and  velocity  of  R  are  equal  to  the 
power  and  velocity  of  Rr,  we  may  write, 


W 


PV 


Wv 
LL^L 

v 


84  TOOTHED   GEARING. 

where  V  and  v  are  the  circumferential  velocities  of  R 
and  r  respectively.  From  this  formula  we  may  write 
the  following :  — 

Ride.  —  The  relative  powers  of  the  wheels  of  a  train 
of  gears  are  inversely  proportional  to  the  circumferen- 
tial velocities  of  the  wheels.  To  find  the  power  of  any 
wheel  of  a  train  of  gears  when  the  power  of  the  next 
wheel  is  known,  multiply  the  power  of  the  latter  by  its 
own  velocity,  and  divide  by  the  velocity  of  the  former. 

Example  i.  —  In  a  train  of  gears  such  as  is  repre- 
sented in  Fig.  76,  the  force  of  the  driver  is  ^==50 
pounds,  the  velocity  of  the  driver  is  V=.  10  feet  per 
second  ;  that  of  the  pulley,  v  =  5  feet  per  second.  Re- 
quired the  weight  W  which  can  be  lifted  by  the  pulley. 
The  force  of  R  is  equal  to  that  of  the  driver,  since  their 
velocities  are  equal.  By  the  rule, 

.  force  of  R'  X  velocity  of  K  _ 

force  of  R  =  -          — = — : ^-~- =  force  of  tf. 

velocity  of  R 

From  formula  (9),  or  the  rule, 

___     PV     50  x  10 

W  =  — •  = =  100  pounds.* 

»  5 

Example  2.  —  In  the  gear  train  represented  in  Fig. 
77  the  force  of  the  driver  R"  is  P=  500  pounds, 
R  =  R'  =  12",  r  —  r'  =  5".  It  is  required  to  find  the 

*  The  gain  in  power  is  obtained  by  a  sacrifice  of  time ;  for  the  wheel 
y?,  having  twice  the  velocity  and  half  the  power  of  the  pulley  r,  can  lift 
twice  as  far  a  weight  equal  to  ^  W  in  the  same  time,  or  just  as  far  a 
weight  equal  to  £  W  in  half  the  time.  The  work  inherent  in  these  two 
wheels  is  therefore  the  same :  r  simply  does  double  work  in  double  time. 
If,  however,  we  have  only  50  pounds  of  force  at  our  disposal,  we  can 
lift  100  pounds  at  one  lift  only  by  means  of  such  a  train,  or  a  similar 
mechanism. 


TOOTHED   GEARING. 


weight,  W,  which  can  be  lifted  by  the  pulley  r,  and  the 
distance  per  minute  which  W  can  be  lifted,  supposing 
the  wheel  R'  to  make  15  revolutions  per  minute.  The 
power  of  R'  is  equal  to  that  of  the  driver  =  P.  From 
formula  (8),  P'  representing  the  power  of  r', 

r>f         pf          n' 
i  J\          * 

~P^V 


12 


=  —  ,     Jr=  1  200  pounds. 
500       5 


This  power  is  transmitted  directly  to  R :  hence 
W     R        W 


P' 


—  — ,     W '  =  2880  pounds. 

1 200  5 


R'  and  r'  make  the  same  number  of  revolutions,  being 

Fig.77 


on  the  same  shaft.     From  formula  (7),  «'  and  n  being 
the  numbers  of  revolutions  of  rr  and  R,  we  have 


r 
R 


15 


12 


The  circumferential  velocity  of  r,  which  is  the  velocity 
with  which  the  weight  W  is  lifted,  is 


2-rrrn 
12 


12 


=  16.36  feet  per  minute. 


86  TOOTHED   GEARING. 

Example  3.  —  Required  an  expression  for  the  weight 
which  can  be  lifted  by  a  train  similar  to  that  of  Fig.  77, 
containing  any  number  of  wheels.  From  Example  2, 

Pr      Rf  „,      PR'        A     W      R  ,,,      P'R 

-rr  =  -r      or    JT  =—  T-    and    -TV  —  -     or     W=  -  . 
P        rr  r'  P'       r  r 

Substituting  in  this  expression  the  value  of  /",  just 

r>  r>  r>/ 

written,  we  obtain  W-=-  —  -7—.     In  the  same  manner, 

rr 

for  any  number  of  wheels,  R,  R  ',  R"  ,  R"'y  etc.,  repre- 
senting the  radii  of  the  large  wheels,  and  ry  r',  r",  /",  etc., 

f  ..  ,  .     1J7      PRR'R"R'",  etc. 

those  of  the  pinions,  we  obtain  W7—  —  __  _____  — 

rrrr,  etc. 

,      D       Wrrr"rm  etc. 
Inversely,  P  = 


Example  4.  —  We  have  a  shaft  which  drives  a  gear 
with  a  force  of  250  pounds  :  we  wish  with  this  power  to 
lift  a  weight  of  1,500  pounds.  Required  the  radii  of  the 
wheels  of  the  necessary  train.  We  can  see  at  a  glance 
that  a  simple  train,  such  as  Fig.  76,  will  not  be1  practi- 

cable, for  in  this  case  —  —  —  —  —  —  =  -  ;  and,  if  r=  6" 

P        r        250        I 

(as  small  as  is  convenient),  R  —  r  X  6  —  36",  or  the 
diameter  of  our  large  gear  will  have  to  be  6  feet.  This 
is  practically  out  of  the  question  :  we  must  therefore 
use  a  train  with  4  or  more  wheels.  Let  us  try  4. 

r)  r>    ir>/ 

From  Example  3  we  have  W=  --  j—.     Taking  r—  r' 


36  36  250 

=  216.     We  can  now  assume  a  value  for  R,  and  find 
the   corresponding  value   of  Rr.      Say  R=.\2"t  then 


TOOTHED    GEARING.  8/ 

In  the  preceding  examples  no  account  has  been  taken 
of  the  friction  of  the  gear  teeth  and  axles,  since  they 
are  given  simply  to  illustrate  the  use  of  the  rules  and 
formulas  which  precede  them.  The  detrimental  fric- 
tion is,  of  course,  very  considerable,  even  in  the  best 
wheels,  and  increases  rapidly  as  we  increase  the  num- 
ber of  wheels  in  a  train  :  therefore  the  trains  spoken 
of  in  the  examples,  if  actually  made  and  used,  would 
accomplish  considerably  less  than  the  examples  give 
them  credit  for.  Were  this  not  the  case,  we  could, 
with  the  slightest  possible  amount  of  power,  by  means 
of  a  train  containing  a  sufficient  number  of  wheels,  per- 
form an  infinitely  great  amount  of  work  —  manifestly, 
from  a  practical  point  of  view  at  least,  an  absurdity. 

§  XII.  —  Line  of  Contact  —  Arc  of  Contact. 

In  a  pair  of  toothed  wheels,  each  tooth  of  one  wheel 
is  in  contact,  for  a  certain,  definite  length  of  time  or 
distance  of  revolution,  with  a  tooth  of  the  other  wheel, 
and  there  is  always  at  least  one  pair  of  teeth  in  contact. 
Whether  or  not  the  same  two  teeth  come  into  contact 
at  each  revolution  depends,  as  we  have  already  seen, 
upon  the  relative  numbers  of  teeth  of  the  two  wheels. 
If,  during  the  contact  of  a  pair  of  teeth,  a  curve  be 
drawn  through  all  the  successive  points  of  contact,  this 
curve  will  represent  the  entire  contact  of  the  teeth. 
Such  a  curve  is  called  the  line  of  contact,  and  its  length 
represents  the  duration  of  the  contact.  The  line  of 
contact  may  be  found  by  drawing  different  positions  of 
two  teeth  while  in  contact,  and  drawing  a  curve  through 
the  points  of  contact  thus  determined.  This  operation 
is,  however,  often  a  difficult  one,  because  the  effect  of 


88 


TOOTHED   GEARING. 


the  preceding  pair  of  teeth  upon  the  early  contact  of 
the  pair  in  question  cannot  easily  be  taken  into  consid- 
eration, and  this  effect  is  very  often  too  important  to 
be  neglected.  Reuleaux  has  pointed  out  the  following 
method  for  determining  the  line  of  contact :  Let  O  and 
O'  (Fig.  78)  be  the  centres  of  two  toothed  wheels  which 
gear  together,  OpO'  the  line  of  centres,  and  /  the  pitch 


point.  From  different  points  along  the  profile  apcf  draw 
normal  lines  intersecting  the  pitch  circle  in  the  points 
b,  b'y  b"  dr,  etc.,  and  from  O  as  a  centre  strike  circle- 
arcs  through  the  points  a,  a',  a",  c,  etc.  We  have  seen, 
that,  for  uniform  velocity  ratio,  it  is  necessary  that  the 
common  normal  to  two  teeth  in  contact  at  the  point  of 
contact  shall  pass  through  the  pitch  point.  If,  there- 
fore, from  the  pitch  point  /  as  a  centre,  with  radii  equal 
to  ab,  a'b ',  a"b",  dc,  etc.,  we  strike  arcs  intersecting  the 


TOOTHED   GEARING.  S() 

above-mentioned  arcs,  the  points  of  intersection  will  be 
points  of  contact  of  the  teeth,  and  a  curve  drawn 
through  these  points  will  be  the  line  of  contact.  The 
arcs  Kp  and  K'p,  taken  on  the  pitch  circles,  and  limited 
by  the  top  circles,  are  called  the  arcs  of  approacJi  and  re- 
cess, according  to  the  direction  of  rotation,  and  together 
form  the  arc  of  contact.  The  length  of  the  arc  of  con- 
tact depends  upon  the  diameters  of  the  pitch  circles  of 
the  gears  and  the  height  of  the  teeth  between  the  pitch 
and  top  circles ;  while  the  length  and  position  of  the  line 
of  contact  depend  not  only  upon  these  dimensions,  but 
also  upon  the  form  of  the  profiles  of  the  teeth  and  the 
number  of  teeth  in  contact  at  one  time.  In  ordinary 
gearing,  where  the  height  of  the  teeth  between  the 
pitch  and  top  circles  is  the  same  for  both  wheels,  the 
arcs  of  approach  and  recess  are  equal,  and,  in  wheels 
having  cycloidal  profiles,  the  lengths  of  the  line  and 
arc  of  contact  are,  according  to  Reuleaux,  equal.  The 
length  of  the  arc  of  contact  must  be  at  least  equal  to 
the  pitch  of  the  teeth,  else  there  would  be  less  than 
one  pair  of  teeth  in  contact  at  one  time  :  in  ordinary 
machine  gearing  this  length  varies  from  i£  to  2|  times 
the  pitch. 

§  XIII.  —  Strength  of  Teeth.  — Rules   for    determining  the  Pitch,  and 
other  Tooth  Dimensions. 

Before  taking  up  the  subject  of  strength  of  wheel 
teeth,  our  notation  for  the  calculations  under  this  head 
must  be  explained.  The  total  height  of  the  tooth, 
i.e.,  the  sum  of  the  heights  above  and  below  the  pitch 
circle,  we  denote  by  //  ( =  //  +  //') ;  the  breadth  of  the 
tooth  on  the  pitch  circle,  by  b ;  and  the  face  width  of 


90  TOOTHED  GEARING. 

the  tooth,  by  /(see  Fig.  79).  In  calculating  the  strength 
of  a  wheel  tooth,  the  curved  profile  is  disregarded,  as  is 
also,  in  ordinary  gearing,  the  influence  of  the  velocity 
of  the  wheel,  and  the  tooth  regarded  as  a  simple  beam 
or  semi-girder  supported  at  one  end,  and  having  a 
weight  or  force,  W,  acting  at  the  other  end  (Fig.  80). 
The  width  b  is  taken  equal  to  the  width  of  the  tooth  on 
the  pitch  circle.  The  safe  working-load  for  a  beam 


Fig.79  Fig.80 


such  as  is  represented  in  Fig.  80  is  expressed  by  the 
formula 

W-^ 
W       6k 

in  which  W  is  the  safe  working-load,  f  the  greatest 
safe  working-stress  in  pounds  per  square  inch  for  the 
material  used,  and  the  other  quantities  the  same  as  in 
Fig.  80.  It  is  evident  that  the  width  b  of  the  tooth 
must  be  less  than  half  the  pitch,  else  the  space  would 
not  be  wide  enough  to  admit  the  tooth  of  the  mate- 
wheel  ;  and,  in  order  that  the  tooth  may  be  sufficiently 
strong  when  it  becomes  worn,  we  take,  at  the  sugges- 
tion of  Unwin,  b  —  Q.^p;  p  being  the  circumferential 
pitch.  Also  we  may  take,  as  is  now  generally  done, 


TOOTHED  GEARING.  9  1 

//  =  //  +  //'  =  o.4/  +  o.  ip  =  o.7/.     These  values,  sub- 
stituted in  the  above  formula,  give 


In  this  expression  W^is  the  actual  load  or  strain  on  one 
tooth.  It  is  more  convenient  to  use  this  formula  in 
terms  of  the  total  force,  P,  transmitted  by  the  wheel. 
Ordinarily,  more  than  one  pair  of  teeth  are  in  gear  at 
once  :  therefore  the  whole  force  transmitted  is  not  sus- 
tained by  one  tooth.  The  number  of  teeth  in  gear  at 
once  varies  considerably  in  different  wheels  ;  but  we 
may  safely  say  that  no  tooth  bears  more  than  three- 
fourths  of  the  entire  force  transmitted.  We  have,  then, 
J'F—  !/>,  and  consequently 

»flx  0.1296^ 


Reducing  this  equation,  we  obtain 
>      0.041  14^»/ 


From  this,  by  transposing, 


p        p 

V  •£_ 

0.04 1 1 4/      / 


or 


This  formula  may  be  termed  the  general  formula  for 
determining  the  pitch.     It  may  be  used  for  any  ma- 


92  TOOTHED   GEARING. 

terial  whatever  by  substituting  for  the  quantity  f  its 
proper  value.  From  the  formula,  therefore,  we  may 
write  the  following  :  — 

General  Rule.  —  To  determine  the  pitch  of  a  gear 
of  any  material,  divide  the  total  force  to  be  transmitted 
by  the  greatest  safe  working-stress  per  square  inch  for 
the  material  of  the  wheel,  multiply  the  quotient  thus 
obtained  by  the  ratio  of  the  pitch  to  the  face  width,* 
extract  the  square  root  of  this  product,  and  multiply 
the  result  by  4.93. 

The  degree  of  safety  necessary  in  calculations  for 
strength  of  gear  teeth  varies  with  the  work  to  be  done 
by  the.  gear,  in  other  words,  with  the  amount  of  clanger 
to  be  incurred.  Thus  the  degree  of  safety  necessary  is 
greater  when  the  gear  is  to  be  subjected  to  sudden, 
violent  shocks  than  when  no  such  shocks  occur,  be- 
cause the  danger  of  breakage  or  accident  is  greater. 
This  degree  of  safety  we  obtain  conveniently  by  vary- 
ing the  value  of  the  quantity  f,  taking  small  Values 
when  the  danger  is  great,  and  vice  versa. 

For  ordinary,  good  cast-iron  we  may  take  f=  4,000 
pounds  when  there  are  sudden,  violent  shocks  upon  the 
gear,  f=  5,000  pounds  when  only  moderate  shocks 
occur,  and  /=  10,000  pounds  when  there  is  little  or 
no  shock.  By  substituting  these  values  of  f,  in  turn, 
in  formula  (10),  and  reducing,  we  obtain  the  following 
formula  for  determining  the  pitch  of  a  cast-iron  gear  :  — 

*  Ordinarily  this   ratio  is   assumed.     For  example,  we  may  assume 


=j  =  - 
sumed. 


-,  or  y  =  -3,  and  determine  the  pitch  for  the  particular  value  as- 


TOOTHED  GEARING. 


93 


For  violent  shock,      /  =  0.078^  P  x  -.     (a) 


For  moderate  shock,/  =  0.07 


For  little  or  no  shock,  /  =  0.05        P  x 


t 


P 


.  —  To  determine  the  pitch  for  a  cast-iron  gear, 
multiply  the  total  force  to  be  transmitted  by  the  ratio 
of  the  pitch  to  the  face  width,  extract  the  square  root  of 
the  product,  and  multiply  the  result  by  0.078  for  violent 
shock,  0.07  for  moderate  shock,  or  0.05  for  little  or  no 
shock. 

In  ordinary   machine-gearing  the  face  width  is  very 

often  taken  equal  to  twice  the  pitch  j/=2/,    y  —  -j; 

because  a  greater  relative  face  width  does  not,  in  the 
same  degree,  add  strength  to  the  tooth,  the  principal 
effect  being  to  increase  the  stiffness  of  the  tooth.  If 

we  make  y  =  i  in  each  of  the  formulas  (u),  we  obtain 


p  =  0.078^  X  |,  /  =  o.07\//'  x  J,   and  /  =  0.05^  Px  ?. 

Reducing  these,   we  have,   for  the  three  cases  given 
above,  the  formulas 


/  =  0.055^ 
/  =  0.05  ^P 
p  =  0.035^ 


(«o 


(0 


(12). 


Rule. — To  determine  the  pitch  for  cast-iron  gears 
when  the  face  width  is  equal  to  twice  the  pitch,  multi- 


94  TOOTHED   GEARING. 

ply  the  square  root  of  the  total  force  to  be  transmitted 
by  0.055  f°r  violent  shock,  0.05  for  moderate  shock,  or 
0.035  f°r  little  or  no  shock. 

A  horsc-poiver,  as  commonly  used,  is  that  force  which 
will  lift  a  weight  of  33,000  pounds  one  foot  high  in  one 
minute,  —  33,000  foot-pounds.  If  we  let  //represent  the 
horse-power,  and  v  the  velocity  at  the  circumference  of 
the  wheel  in  feet  per  second,  we  shall  have  the  expres- 
sion, 

_  550^ 


607;  v 


This  value  of  P,  substituted  in  formulas  (11),  gives  the 
following  convenient  formulas  for  the  pitch  when  the 
horse-power  and  the  velocity  in  feet  per  second,  instead 
of  the  force  transmitted  in  pounds,  are  given  :  — 

For  violent  shock,        p  =  1&Z\  ~~  7     (a) 

I  j-f  -ft 
For  moderate  shock,   /  =  i.64\/ —  -j     (b) 

For  little  or  no  shock,  p  —  i.i  7y  —  -     (c) 

Rule.  —  To  determine  the  pitch  from  the  horse- 
power and  velocity  in  feet  per  second,  multiply  the 
ratio  of  the  horse-power  to  the  velocity  by  the  ratio  of 
the  pitch  to  the  face  width,  extract  the  square  root 
of  the  product,  and  multiply  the  result  by  1.83  for 
violent  shock,  1.64  for  moderate  shock,  or  1.17  for 
little  or  no  shock. 

By  substituting  the  above  value  of  P  in  formulas  (12), 
we  obtain  for  the  pitch,  when  the  face  width  is  not  less 
than  twice  the  pitch,  the  formulas  :  — 


TOOTHED   GEARING. 


95 


For  violent  shock,       p 


'  Jl 

For  moderate  shock,  /  =  i.iyV/  —     (b) 

For  little  or  no  shock,  /  =  0.8 2V/  —      (<r) 

vv 


(14) 


Rule.  —  To  determine  the  pitch  from  the  horse- 
power and  velocity  in  feet  per  second,  when  the  face 
width  is  equal  to  twice  the  pitch,  multiply  the  square 
root  of  the  ratio  of  the  horse-power  to  the  velocity  by 
1.29  for  violent  shock,  1.17  for  moderate  shock,  or  0.82 
for  little  or  no  shock. 

If  we  represent  by  ;/  the  number  of  revolutions  per 
minute,  and  by  D  the  diameter  of  the  wheel  in  inches, 
we  may  obtain,  for  the  velocity  in  feet  per  second,  the 
value, 


v  = 


12    X    60 


=  0.0043  6-Z?». 


This  value,  substituted  for  v  in  formulas  (13),  gives 
For  violent  shock, 
For  moderate  shock,  /  =  24.84X7  yr-  j     (b) 

I      TT        L 

For  little  or  no  shock,  /  =  1 7-72\/ 77-  f     (0 

Rule.  —  To  determine  the  pitch  from  the  horse- 
power, diameter,  and  number  of  revolutions,  divide  the 
horse-power  by  the  product  of  the  diameter,  in  inches, 
into  the  number  of  revolutions  per  minute,  multiply 
the  quotient  by  the  ratio  of  the  pitch  to  the  face  width, 


96  TOOTHED   GEARING. 

extract  the  square  root  of  the  product,  and  multiply  the 
result  by  27.71  for  violent  shock,  24.84  for  moderate 
shock,  or  17.72  for  little  or  no  shock. 

By  substituting  v  =  o.oo^6Dn  for  v  in  formulas  (14), 
the  following  formulas  for  the  pitch,  when  the  face 
width  is  equal  to  twice  the  pitch,  may  be  obtained  :  — 


For  violent  shock 


/  // 

,      /  =  I9.54y  2^     (a) 


-=- 


(*) 


For  moderate  shock,/  =  i7.72y 

I~S" 
For  little  or  no  shock,/  =  1 2.42V/ -=r-     (t) 


06). 


Rule. — To  determine  the  pitch  from  the  horse-power, 
diameter,  and  number  of  revolutions,  when  the  face 
width  is  equal  to  twice  the  pitch,  divide  the  horse- 
power by  the  product  of  the  diameter  into  the  number 
of  revolutions,  extract  the  square  root  of  the  quotient, 
and  multiply  the  result  by  19.54  for  violent  shock;  17.72 
for  moderate  shock,  or  12.42  for  little  or  no  shock. 

Example  I.  —  A  cast-steel  gear  wheel  is  required 
which  will  transmit  a  force  of  100,000  pounds.  The 
gear  is  to  be  subjected  to  severe  shock.  Suppose  a 
specimen  of  the  steel  to  be  used  has  been  tested,  and 
the  breaking-weight  found  to  be  140,000  pounds  per 
square  inch.  We  may  take,  for  the  greatest  safe  work- 
ing-stress per  square  inch,  /  =  \  X  140000  =  23333 

pounds,  say,  /•=.  23000.     If,  now,  we  take    ^  =  -,  for- 

/        4 

mula  (10)  gives, 


=  4.93V/^^X-  =  4-93V/T^  =  4-93  X  1.042  =  5.137 
^V   23000      4 


n. 


TOOTHED   GEARING.  97 

For  the  other  dimensions  we  have 

/  =  4/  =  4  X  5.137  =  20.548  inches 

h  =  Q.ip  =  0.7  x  5.137  =  3.5959  inches 
and 

b  =  0.46^  =  0.46  x  5.137  =  2.363  inches. 

Taking  these  values  to  the  nearest  eighth,  sixteenth, 
etc.,  taking  care  to  err  only  on  the  safe  side,  we  have 
/  =  5  6i  inches,  /  =  2OT9g  inches,  //  =  3  if  inches,  and 
b  =  2  J  inches. 

Example  2.  —  Required  the  tooth  dimensions  for  a 
wooden  cog-wheel  to  transmit  a  force  of  10,000  pounds, 
moderate  shock.  Let  us  suppose  the  cogs  to  be  of 
oak,  the  breaking-strength  of  which  is  15,000  pounds 
per  square  inch.  We  may  safely  take 

/=  \  X  15000  =  2500  pounds. 
If  j  =  -,  formula  (10)  gives 


=  4-93^/ 


IOOOO          I  .— 

X  -  =  4-93V2  =  4-93  X  1.414  =  6.971  m. 


/=  2p  =  13.942  inches 

h  =  0.7^  =  4.8797  inches 
and 

b  =  0.46^  =  3.20666  inches. 

In    fractions    the    dimensions    are,    /  =  6\\    inches, 
/=  i3|-J  inches,  h  =  4^  inches,  and  d=  3^2  inches. 

Example  3.  —  With  the  data  of  Example  2,  required 
the  tooth  dimensions  for  a  wheel  to  be  subjected  to 


98  TOOTHED   GEARING. 

little  or  no  shock.     In  this  case  we  may  take 

/=  i  x  15000  =  3750. 
As   before,  formula  (10)  gives  for  the  pitch, 


P  =  4'9  X      =  4*93V/I'333  =  4-93  X  1.154  =  5-689 


k  =  o.7/  =  3.982" 
b  =  0.46^=  2.617". 

Example  4.  —  Required  the  tooth  dimensions  for  a 
cast-iron  gear  wheel  which  is  to  be  worked  by  hand 
and  crank.  Suppose  a  man  can  exert  by  means  of  the 
crank  a  force  of  300  pounds  on  the  wheel.  From 
formula  (12,  c)  we  have,  for  little  or  no  shock, 

p  =  0.035^300  =  0.035  x  I7-32  =  0-606" 
/=  2/=  1.212" 

^  =  0.7^  =  0.4242" 
and 

£  =  0.46^  =  0.37876". 

If,  for  any  reason,  it  is  necessary  to  make  7=-,  we 

/      4 

must  use  formula  (u,  c),  which  gives 

p  =  0.05^300  x  J  =  0.05  x  8.66  =  0.433". 

Hence 

/=4/=  1.732",  etc. 

Example  5.  —  A  cast-iron  gear  is  to  have  a  circumfer- 


TOOTHED   GEARING.  99 

ential  velocity  of  10  feet  per  second,  and  is  to  transmit 
a  force  of  15-horse  power,  moderate  shock.      Required 

the  dimensions  of  the  teeth  when  -,-=. — .     From  for- 

/      3i 
mula  (13,  b), 


p  =  i.64         x-    =  1.64^0.4286  =  1.64x0.6546  =  1.0725 
Hence 


",  etc. 
It  j  =  -,  formula  (14,  b)  gives 

p  =  ItX  Vxo  =  '-'f^S0*  =  l'^  x  x 


xo  =  1-43*9" 

and 

/=  2p  =  2.8658",  etc. 

Example  6.  —  The  diameter  of  a  cast-iron  gear  which 
is  to  transmit  25-horse  power,  excessive  shock,  is  30". 

Required  the  dimensions,  taking  "?  =  -»  and  the  number 
of  revolutions  per  minute  ;/  =  80.    From  formula  (i6,#), 


=  19-54^0.0104  =  19.54  X  0.102  =  2" 


/  =  4",etc. 

Example  J.  —  Required  a  cast-steel  gear  which  will 
safely  transmit  loo-horse  power,  excessive  shock.  The 
diameter  of  the  gear  is  to  be  36",  and  the  number  of 
revolutions  per  minute  20.  From  the  expressions 


100  TOOTHED   GEARING. 

v  —  0.00436/7/2  and  P  =  —  -  we  have,  by  combining, 

v 

cco/7        550  x  100 

P  =   JJ  .  _  =  -  -  =  17520  pounds. 

0.00436  X  36  X  20 


Taking  f=  23,000  pounds,  as  in  Example  i,  and  7  =  -, 

/      4 

formula  (10)  gives 


x  7  =  4.93^0.19  =  4-93  X  0.436  =  2.149" 
23000       4 

/  =  4/  =  8.596",  etc. 

As  has  already  been  noticed,  the  ratio  ^  is  com- 
monly taken  equal  to  J ;  that  is,  the  face  width  equals 
twice  the  circumferential  pitch.  Formulas  (12),  (14), 
and  (16),  which  have  been  obtained  from  formulas  (11), 

(13),  and  (15)  by  substituting  for    -  the  value  |,  are 

therefore  much  the  more  often  used.  The  following 
tables,  obtained  from  formulas  (12),  (14),  and  (16),  will 
be  found  convenient :  — 


TOOTHED   GEARING. 


101 


TABLE   I. 

From  formula  (12,  a,  b,  and  c). 


p 

in  inches. 

7>for 

No. 

Little  or  no  shock. 

Moderate  shock. 

Violent  shock. 

r 

816 

4OO 

331 

, 

l| 

1,276 

625 

513 

2 

'* 

1,837 

900 

743 

3 

If 

2,500 

1,225 

1,012 

4 

2 

3,265 

1,  6OO 

1,322 

5 

2* 

4J33 

2,025 

1,670 

6 

2£ 

5,102 

2,500 

2,066 

7 

2f 

6,173 

3,025 

2,5OO 

8 

3 

7,347 

3,600 

2,975 

9 

31 

8,622 

4,225 

3,49i 

10 

3£ 

10,000 

4,900 

4,050 

n 

sf 

11,480 

5,625 

4,649 

12 

4 

13,061 

6,400 

5,289 

13 

41 

14,745 

7,225 

5,97i 

14 

4* 

16,531 

8,100 

6,694 

15 

4f 

18,418 

9,025 

7,459 

16 

"5 

20,408 

10,000 

8,265 

17 

5i 

22,500 

11,025 

9,111 

18 

5£ 

24,694 

12,100 

10,000 

19 

Sf 

26,990 

13,225 

10,930 

20 

6 

29,388 

14,400 

11,900 

21 

6£ 

34,490 

16,900 

13,967 

22 

7 

40,000 

19,600 

16,198 

23 

7* 

45,918 

22,500 

i8,595 

24 

8 

52,245 

25,600 

21,157 

25 

102 


TOOTHED   GEARIfiG. 


TABLE   II. 

From  formula  (14,  #,  b,  and  c.) 


p 

in  inches. 

H  , 

-  for 
v 

No. 

Little  or  no  shock. 

Moderate  shock. 

Violent  shock. 

I 

1.49 

0-73 

O.6O 

I 

'* 

2.32 

1.  14 

0.94 

2 

«* 

3-35 

1.64 

1-35 

3 

I42 

4-55 

2.24 

1.84 

4 

2 

5-95 

2.92 

2.40 

5 

2* 

7-53 

3.69 

3.04 

6 

2£ 

9-30 

4.56 

3.76 

7 

2| 

11.25 

5.52 

4-54 

8 

3 

13-38 

6-57 

541 

9 

3* 

15.71 

7.72 

6-35 

10 

3£ 

18.22 

8.95 

7.36 

ii 

si 

20.91 

IO.27 

8.45 

'     12 

4 

23.80 

11.69 

9.61 

13 

4* 

26.86 

I3-I9 

10.85 

H 

4£ 

30.11 

14.79 

12.11 

*5 

4f 

33-56 

16.48 

I3.56 

16 

5 

3M8 

18.26 

15.02 

17 

5* 

40.99 

20.13 

16.56 

18 

5£ 

44.99 

22.09 

18.18 

19 

5f 

49.17 

24.15 

19.87                       20 

6 

53-54 

26.30 

21.63                       2I 

6* 

62.83 

30.86 

25-39 

22 

7 

72.87 

35.80 

29.45 

23 

7£ 

83.66 

41.09 

33-80 

24 

8 

95.18 

46.75 

38.46 

25 

TOOTHED   GEARING. 


TABLE  III. 
From  formula  (16,  #,  <5,  and  c). 


p 

in  inches. 

H 
Ti~  for 

z># 

No. 

Little  or  no  shock. 

Moderate  shock. 

Violent  shock. 

I 

0.0065 

0.0032 

O.OO26 

I 

I* 

O.OIOI 

0.0050 

O.004I 

2 

I* 

0.0146 

0.0072 

O.OO59 

3 

If 

0.0198 

0.0098 

O.OO8O 

4 

2 

0.0259 

0.0127 

O.OIO5 

5 

2* 

0.0328 

0.0161 

0.0133 

6 

2* 

0.0405 

0.0199 

0.0164 

7 

2j 

0.0490 

O.024I 

0.0198 

8 

3 

0.0583 

0.0287 

0.0236 

9 

31 

0.0685 

0.0336 

0.0277 

10 

3£ 

0.0794 

0.0390 

0.0321 

ii 

3! 

0.0912 

0.0448 

0.0368 

12 

4 

0.1037 

0.05IO 

0.0419 

13 

4J 

0.1171 

0.0575 

0.0473 

H 

4£ 

0.1313 

0.0645 

0.0530 

15 

.    4f 

0.1463 

0.0719 

0.0591 

16 

5 

0.1621 

0.0796 

0.0655 

17 

Si 

0.1787 

0.0878 

0.0722 

18 

5* 

0.1961 

0.0963 

0.0792 

19 

si 

0.2143 

0.1053 

0.0866 

20 

6 

0.2334 

O.II46 

0.0943 

21 

^ 

0.2739 

0.1346 

O.II07 

22 

7 

0.3177 

0.1560 

0.1283 

23 

7* 

0.3647 

0.1790 

0.1473 

24 

8 

0.4149 

0.2038 

0.1676 

25 

IO4  TOOTHED   GEARING. 

Example  I. —  Required  the  pitch  of  a  cast-iron  bevel 
wheel  which  will  transmit  a  force  of  10,000  pounds, 
moderate  shock.  In  Table  L,  column  for  moderate 
shock,  line  17,  we  find  P  =  10,000  pounds.  In  the  pitch 
column,  and  directly  opposite  this  value  of  P,  we  find 
the  required  pitch,  /  =  5".  Hence  /=  2p  —  10",  etc. 

Example  2. — The  force  transmitted  by  a  cast-iron 
gear  under  violent  shock  is  6,000  pounds.  Required 
the  necessary  pitch.  Table  L,  column  for  violent  shock, 
line  14,  gives  P  =  5,971  pounds  ;  and  the  corresponding 
pitch  is  /  —  4j".  Since  this  pitch  corresponds  to  a 
value  of  P  slightly  less  than  the  required  one,  we  may 
take  for  our  required  pitch  /  =  4§". 

Example  3.  — The  pitch  of  a  cast-iron  gear  subjected 
to  little  or  no  shock  is  2^".  Required  the  force  in 
pounds  which  can  be  safely  transmitted  by  the  gear. 
In  Table  L,  pitch  column,  line  6,  we  find/  =  2%'.  The 
value  of  P  for  little  or  no  shock,  corresponding  to  this 
pitch,  is  4,133  pounds. 

Example  4.  —  Required  the  pitch  for  a  cast-iron  gear 
which  will  safely  transmit  24-horse  power,  violent 
shock,  at  a  circumferential  velocity  of  8  feet  per  sec- 
ond. In  this  case  —  =  —  =  3.  In  Table  II.,  column 
v  8 

TT 

for  violent   shock,  line  6,  we  find  —  =  3.04 ;  and  the 

TT 

corresponding  pitch   (found  opposite   this  value  of  — 

V 

in  the  pitch  column)  is  p  —  2 J". 

Example  5. — A  certain  cast-iron  gear  transmits  75- 
horse  power.  The  pitch  of  the  gear  is  3|".  Required 
the  circumferential  velocity  safe  for  the  gear  at  mod- 


TOOTHED   GEARING.  1  05 

erate   shock.     We   have   from   Table   II.,    column   for 

TT 

moderate  shock,  the  value  —  —8.95,  corresponding  to 

/=3j.      Hence  ^^8.95,     v  =  ^=  8.38  feet  per 
v  8.95 

second. 

Example  6.  —  Required  the  pitch  for  a  cast-iron  gear 
to  transmit  safely  5O-horse  power,  violent  shock,  at  100 
revolutions  per  minute  ;  the  diameter  of  the  gear  being 
16".  We  have 

H  50  i 


In  Table  III.,  column  for  violent  shock,  line  11,  we  find 

TT 

—  -  =  0.0321.     The  corresponding  pitch  is/  =  3^". 

The  following  table  will  be  found  very  convenient  in 
converting'decimals  into  fractions:  — 


io6 


TOOTHED   GEARING. 
TABLE   IV. 


V) 

<n 

ui 

tn 

t/i 

-/-: 

M 

• 

_c 

tj. 

00 

vO 

*w 

^J. 

"-J. 

00 

•O 

i 

3. 

CO 

VO 

" 

m 

VO 

I 

0.015625 

33 

0.515625 

1 

0.031250 

17 

0.531250 

3 

0.046875 

35 

0.546875 

I 

0.062500 

9 

0.562500 

5 

0.078125 

37 

0.578125 

3 

0.093750 

19 

o.59375o 

7 

0.109375 

39 

0.609375 

I 

0.125000 

5 

0.625000 

9 

0.140625 

41 

0.640625 

5 

0.156250 

21 

0.656250 

ii 

0.171875 

43 

0.671875 

3 

0.187500 

ii 

0.687500 

13 

0.203125 

45 

0.703125 

7 

0.218750 

23 

0.718750 

15 

0.234375 

47 

0-734375 

I 

O.25OOOO 

3 

0.750000 

17 

0.265625 

49 

0.765625 

9 

0.281250 

25 

0.781250 

19 

0.296875 

51 

0.796875 

5 

0.312500 

13 

0.812500 

21 

0.328125 

53 

0.828125 

ii 

0.343750 

27 

0.843750 

23 

0-359375 

55 

0.859375 

3 

0.375000 

7 

0.875000 

25 

0.390625 

57 

0.890625 

13 

0.406250 

29 

0.906250 

27 

0.421875 

59 

0.921875 

7 

0.437500 

15 

0.937500 

29 

0.453125 

61 

0.953125 

15 

0.468750 

31 

0.968750 

31 

0.484375 

63 

0.984375 

2 

0.500000 

4 

1.  000000 

TOOTHED   GEARING.  IO? 

For  very  high  speed  gears,  we  may  take,  at  the  sug 
gestion  of  Professor  Reuleaux  and  Mr.  W.  C.  Unwin, 


f=-     —  ,  v  being  the  circumferential  velocity  in  feet 

•Jv 
per  second.     For  example,  suppose  it  was  required  to 

determine  the  pitch  of  a  cast-iron  gear,  the  velocity  of 
which  is  35  feet  per  second,  and  the  force  to  be  trans- 
mitted is  5,000  pounds.  We  have  for  our  safe  working- 
stress 

.         I  0000          I  0000 


From  formula  (10),  taking  *-~  =  -,  we  have 

/       4 


/CQOO  I  /  -  • 

=  4'93V  3058  X  4  =  4-93Y0.4088  =  4.93  x  0.639 


§  XIV.  —  Strength  of  Arms,  Rim,  Nave,  Shafts,  etc. 

The  arms  of  a  gear-wheel  being  symmetrically  placed 
with  reference  to  each  other  and  to  the  rim  of  the 
wheel,  we  may  assume,  without  demonstration,  that 
each  arm  bears  an  equal  share  of  the  total  strain  upon 
the  rim  :  in  other  words,  the  strain  upon  each  arm  is 
equal  to  the  total  strain  upon  the  rim  divided  by  the 
number  of  arms  in  the  gear.  This  is  a  bending-strain 
acting  in  a  direction  perpendicular  to  the  axis  of  the 
arm  and  in  the  plane  of  the  wheel.  The  proper  strength 
for  the  arms  may  therefore  be  calculated  similarly  to 
that  of  the  teeth. 

Fig.  8  1  represents  a  portion  of  a  gear  showing  one 


io8 


TOOTHED   GEARING. 


arm.  R  is  the  radius  of  the  pitch  circle  in  inches  ;  P, 
the  total  strain  upon  the  rim  (the  total  force  transmit- 
ted in  pounds)  ;  //,  and  /;t  respectively,  the  width  (in  the 
plane  of  the  wheel)  and  thickness  of  the  arm  in  inches  ; 
and  ;//,  the  number  of  arms.*  Denoting  by  /  the 
greatest  safe  working-stress  for 
the  material,  the  equation  for 
equilibrium  is 

P  _  fbji? 


Fig. 81 


We  may  take,  in  this  case,  for 
cast-iron,  /  =  3,000  pounds.  Con- 
sequently 

P 


P  »/  6R 

From  this,  by  transposing,  we  have 


which  may  be  easily  solved  by  assuming  a  value  for  bl 

in  terms  of  //,  {  —  =  -,     -r  =  ->  etc.),  and  finding  the 
\/tl      2       ftt      4         / 

corresponding  value  of  hv 

For  convenience,  we  may  write  formula  (17)  in  the 
form  of  an  equation  having  one  unknown  quantity, 
thus : — 

(18) 


*  The  dimensions  bi  and  7/x  are  taken  at  the  rim  of  the  wheel,  and 
tapered,  as  shown  in  Fig.  81. 


TOOTHED   GEARING. 


109 


and  find  values  of  the  co-efficient  x  for  different  values 
of  bt  and  «/.* 

The  following  table  gives  values  of  x  for  different 

values  of  -^  and  «/  :  — 
h* 

TABLE   V. 


1 

;rfor  «x'= 

k\ 

4 

5 

6 

8 

IO 

i 

O.I  00 

0.093 

0.087 

0.079 

0.074 

1 

0.126 

0.117 

O.IIO 

O.I  00 

0.093 

1 

0.144 

0.134 

0.126 

0.114 

O.I  06 

1 

0.159 

0.147 

0.139 

0.126 

0.117 

Rule.  —  To  determine  the  width  of  cast-iron  gear 
arms  in  the  plane  of  the  wheel,  multiply  the  force 
transmitted  by  the  radius  of  the  pitch  circle,  extract 
the  cube  root  of  this  product,  and  multiply  the  result 
by  the  tabular  number  corresponding  to  the  given  values 

of  ^  and  »/. 
h, 

Example  i.  —  A  cast-iron  gear  the  diameter  of  which 
is  48"  transmits  a  force  of  5,000  pounds.  Required 
the  width  and  thickness  of  the  arms,  of  which  there  are 

5.     If  we  assume  71  =  -,  Table  V.  gives,  for  the  value 


of    the   co-efficient,  x  =.  o.  1  1  7. 
becomes 


Hence   formula   (18) 


*  This  form  is  given  by  Umvin  in  Elements  of  Machine  Design. 


10  TOOTHED   GEARING. 


hl  =  o.i  171/7^  =  o.uysooo  x  24  =  o.ny'  12000  =  0.117 

X  49-324  =  5-77" 
or  in  fractions,  from  Table  IV.,  //,  =  5ff":  hence 

*,  =  #,  =  i  X  5-77  =  14425"  =  i&". 

Example  2.  —  A  cast-iron  72"  gear  transmits  a  force 
of  15,000  pounds.     Taking  n'  =  6,  and  —  -  —  -,  required 

//!  2 

the  dimensions  of  the  arms.    From  Table  V.,  x  =•  0.087  : 
hence,  from  formula  (18),  we  have 


hl  =  0.08  7  V^#  =  0.08 7'V 1 5 ooo  x  36  =  0.087  X  81.433 

=  7-0847"  =7*". 
For  the  thickness  we  have 

*,  =  fa  =  j  x  7.0847  =  3-54235"  =  3H"-    - 

If,  instead  of  rectangular,  we  have  circular  cross- 
sections  for  gear  arms,  and  represent  the  diameter  by 
dr,  the  equation  for  equilibrium  becomes 

P  _  /x  0.0982^3 
«?"        ~~R~ 
or,  for  cast-iron, 

P  _  3000  x  0.0982^'* 
«7~          ^ 

Reducing  and  transposing  this  equation  gives 

PR 


TOOTHED   GEARING.  Ill 

Rule.  —  To  determine  the  diameter  for  cast-iron  gear 
arms  having  circular  cross-sections,  multiply  the  force 
transmitted  by  the  pitch  radius,  divide  this  product  by 
the  number  of  arms,  extract  the  cube  root  of  the  quo- 
tient thus  obtained,  and  multiply  the  result  by  0.15. 

Example  3. — A  cast-iron  gear  of  $6"  diameter  has 
5  arms  (circular  cross-sections),  and  transmits  a  force 
of  600  pounds.  Required  the  diameter  for  the  arms. 
From  formula  (19)  we  have 


,,  *  3/i  8  X  600  R/  —  -—  „ 

d  =  o.i5y  ---  =  o.i5V2i6o  =  0.15  x  12.927  =  1.939 

or,  from  Table  IV.,  </'=  i^£". 

For  elliptical  cross-sections,  representing  by  a  the 
major  and  by  b'  the  minor  axis,  the  equation  for  equilib- 
rium is 

P  _  f  x  0.0982^2 
_____         _____ 

or,  for  cast-iron, 

P  _  3000  X  0.0982^2 


Hence,  by  reducing  and  transposing,  we  obtain 

PR 

0.00339—7-        (20). 


Rule.  —  To  determine  the  dimensions  for  cast-iron 
gear  arms  having  elliptical  cross-sections,  multiply  the 
force  transmitted  by  the  pitch  radius,  multiply  the  prod- 
uct thus  obtained  by  0.00339,  and  divide  the  result  by 
the  number  of  arms.  This  gives  the  product  of  the 


112  TOOTHED   GEARING. 

minor  into  the  square  of  the  major  axis  (b'a2)  :  the  axes 
may  then  be  found  as  in  formula  (17). 

Example  4.  —  Required  the  axes  for  the  cross-sections 
of  the  elliptical  arms  of  a  cast-iron  gear,  the  diameter 
of  which  is  24".  The  force  transmitted  is  800  pounds, 
and  the  number  of  arms  3.  Let  us  assume  a  relation 

between  the  cross-section  axes,  say  £'  =  -.     Formula 
(20)  then  gives 


a*  800  x  12 

or 


-  =  0.00339 


Q-00339  X  800  x  12  X  2  = 
3 


Hence 

a  =  V  21.  696 

=  2.789"  =2 

Also 

j,=  g  =  g-y89 

2  ~~          2 

=  1-3945"  = 

For  arms  having  flanged  cross-sections,  such  as  is 
shown  in  Fig.  82,  the  equation  for  equilibrium  becomes 

'*  4-  BhJ 


-7  =  ^X 

7/t          A  ujj 

Substituting  for  /  its  value  of  3,000  pounds,  and  re- 
ducing, we  obtain,  for  cast-iron, 

P  _  5oo(£,,//'3  4- 
*7~  RH' 

or 

-  Bh,t         PR 


TOOTHED   GEARING. 


Example  5.  —  Required  the  dimensions  for  the  arms 
of  a  36"  cast-iron  gear  which  transmits  a  force  of  800 
pounds  ;  the  arms  to  be  flanged,  as  in  Fig.  82,  and  to 
be  4  in  number.  Let  us  assume  relations  between  the 


Fig.82 


Fig. 83 


several  unknown   quantities   in   formula   (21).     Thus, 
suppose 

h       J,       B     H' 
b,,=  h,,=  -~—. 

By  substitution  the  formula  becomes 


800  X  18 

5°°  *  4  ' 


H' 


Reducing,  we  have 


H'*         20'*  _   I  2  iff'*  _  $6 

5        625          625       '  5* 


Hence 


5X127 


114  TOOTHED   GEARING. 

For  the  other  dimensions, 


and 

B-tZ. -'-*&*£- w&. 

Converted  into  fractions  by  means  of  Table  IV.,  the  arm 
dimensions  are  H'  =  3^2 ">  b,,  =  /i,,=  fi",  and  #~  iT5g". 
For  arms  with  cross-sections,  flanged  as  in  Fig.  83, 
the  equation  for  equilibrium  is 

P___f_      BIT*  -  b,,h,* 
n/  ~~  7?  X  6H'        ' 

=  3,000  in  this  equation,  we  have,  for  cast-iron, 
P 


(22)* 


n!~  RH' 

or 

b,,h,t        PR 


Example  6.  —  A  48"  cast-iron  gear  transmits  a  force 
of  i ,000  pounds,  and  has  5  arms,  the  cross-sections 
being  flanged  as  in  Fig.  83.  Required  the  arm  dimenr 

TTf  J 

sions.     Let  us  take  B=—t  ////  =  £//',  and  —~\H'. 

2  2 

These  values,  substituted  in  formula  (22),  give 

1000  X  24 
H1  5°°  X  5  * 

Reducing,  we  have 


256 


TOOTHED   GEARING. 


and 

=  0.3623"  = 


The  number  of  arms  in  a  gear-wheel  is  often  deter- 
mined, according  to  the  pitch  diameter,  by  the  following 
table  :  — 

For  a  gear  of  i  J  to  3^  feet  diameter,  4  arms. 
For  a  gear  of  3^  to  5  feet  diameter,  5  arms. 
For  a  gear  of  5  to  8£  feet  diameter,  6  arms. 
For  a  gear  of  8|  to  16  feet  diameter,  8  arms. 
For  a  gear  of  16  to  25  feet  diameter,  10  arms. 

Reuleaux  gives  for  the  number  of  arms  the  formula 

(23) 

in  which  ;//  is  the  number  of  arms,  ^V  the  number  of 
teeth  in  the  gear,  and/  the  pitch.* 

Ride.  — To  determine  the  number  of  arms  for  a  gear- 
wheel, extract  the  square  root  of  the  number  of  teeth 
and  the  fourth  root  of  the  pitch,  multiply  the  roots 
together  and  the  product  by  0.56. 

Example  6  a. — A  gear-wheel  has    100  teeth  and  a 

*  Small  pinions,  and  sometimes  narrow-faced  gears,  are  made  without 
arms;  i.e.,  having  a  continuous  web  cast  between  the  rim  and  nave. 


Il6  TOOTHED   GEARING. 

pitch  of  i".     Required  the  number  of  arms.     From  for 
mula  (23) 

«/  =  0.56^100  V7  =  0.56  X  10  X  i  =  5.6  or  6. 

A  convenient  formula  for  the  arm  dimensions,  in 
terms  of  the  horse-power  transmitted  and  the  revolu- 
tions, may  be  obtained  as  follows.  As  explained  in 
§  XIII.,  we  have  the  expressions 


v  =  0.008     Rn    and    P= 


v  being  the  circumferential  velocity  in  feet  per  second, 
H  the  horse-power,  and  n  the  revolutions  per  minute. 
By  combining  these  we  obtain 


H 


This  value  of  P  substituted  in  formula  (17)  give's 
63000;?        R 

€s  i  *fr  i      •—  —  n  s\  t 

Rn          SOCK/ 
or 

IT 

M,a=i26—  ,         (24). 
Tin  i 

Rule.  —  To  determine  the  quantity  bji?  (the  thick- 
ness multiplied  by  the  square  of  the  width)  for  cast-iron 
gear  arms,  from  the  horse-power  and  revolutions,  mul- 
tiply the  horse-power  by  126,  and  divide  by  the  product 
of  the  number  of  revolutions  into  the  number  of  arms. 

Example  7.  —  A  36"  cast-iron  gear  makes  80  revolu- 
tions per  minute,  and  transmits  15  -horse  power.  Re- 


TOOTHED   GEARING.  1 1/ 

quired  the  dimensions  of  the  arms.  From  the  table  we 
have  for  the  number  of  arms  ;//  =  4,  and  from  for- 
mula (24) 

126  x  15 

^ = -8^r :=  s-9°6- 

We  may  now  assume  bl  =  — •  :  hence 

7/3 
Mi2  =  —  =  5-906 

h*  =  ^23.624  =  2.869"  =  2J" 
and 

_/Zr_   2.869  _  23" 

*,----  -—     :  0.717 

For  arms  having  circular  cross-sections  we  have,  as 
above, 

P=  63000^ 

which,  substituted  in  formula  (19),  gives,  for  the  diame- 
ter of  the  arm  cross-section, 


or  

'  Hr,        (25). 


Rule.  —  To  determine  the  diameter  for  cast-iron  gear 
arms  having  circular  cross-sections,  from  the  horse- 
power and  revolutions,  divide  the  horse-power  by  the 
product  of  the  number  of  revolutions  per  minute  into 
the  number  of  arms,  extract  the  cube  root  of  this  quo- 
tient, and  multiply  the  result  by  5.969. 

Example  8.  —  The  diameter  of  a  cast-iron  gear  is  48", 


Il8  TOOTHED   GEARING. 

the  horse-power  transmitted  15,  and  the  number  of  revo- 
lutions per  minute  40.  Required  the  diameter  for  the 
circular  cross-sections  of  the  arms.  From  the  table, 
the  number  of  arms  is  5  :  hence,  from  formula  (25), 


5-969  X  0.4217  =  2.517  =  2ff". 


For  elliptical  cross-sections,  of  which  a  and  b'  are 
respectively  the  major  and  minor  axes,  we  have,  by 
substituting  in  formula  (20),  the  value 

/>=  63000  -g, 

H      R 

b'a?  =  0.00339  x  63000  •=-  x  — > 
Rn      n' 

or 

*V  =  2I3-57J?     (26)- 

Rule. — To  determine  the  quantity  b'a?  (the  minor 
axis  multiplied  by  the  square  of  the  major),  for  cast-iron 
gear  arms  having  elliptical  cross-sections,  from  the 
horse-power  and  revolutions,  multiply  the  horse-power 
by  213.57,  and  divide  by  the  product  of  the  number  of 
revolutions  per  minute  into  the  number  of  arms. 

Example  9.  —  A  48"  cast-iron  gear  makes  40  revolu- 
tions per  minute,  and  transmits  2O-horse  power.  Re- 
quired the  arm  dimensions  for  elliptical  cross-sections. 
In  this  case,  n'  =  4,  and  hence  formula  (26)  gives 


TOOTHED   GEARING.  119 

If  we  take  b'  =  \a,  we  shall  have 


a* 
*  =  --  =  21.357 


«  =    ai.357X  2  =  3496"=  3*" 
and 


For  arms  having  cross-sections  flanged,  as  shown  in 
Fig.  82,  we  obtain,  by  substituting  in  formula  (21)  the 
value  of  P  determined  above, 

b.,H'*  +  Bh,t  H          R 

-  —  -  --  =  63000  -5-  X  --  7 
H'  Rn      soo«, 

or 


(27) 


which  may  be  solved  as  explained  in  Example  5  of  this 
section. 

Similarly,  for  arms  having  cross-sections  flanged,  as 
in  Fig.  83,  we  obtain 

(28)* 


It  is  often  convenient  to  calculate  the  dimensions 
of  the  arms  from  the  pitch  and  radius  of  the  gear. 
Formulas  for  the  arm  dimensions,  in  terms  of  these 
quantities,  may  be  obtained  as  follows  :  — 

From  formula  (12,  b)  we  may  write 

~"  0.0025 


I2O  TOOTHED   GEARING. 

which,  substituted  in  formula  (17),  gives 


or 


?.  —  To  determine  the  quantity  bji?  (the  thick- 
ness of  the  arm  multiplied  by  the  square  of  its  width) 
from  the  pitch  and  radius  of  the  gear,  divide  the  con 
tinned  product  of  0.8  into  the  square  of  the  pitch  into 
the  radius,  by  the  number  of  arms. 

Example  10.  —  Required  the  dimensions  for  the  arms 
of  a  gear-wheel,  the  diameter  of  which  is  24",  and  the 
pitch  i".  In  this  case,  n^-=.^\  hence,  from  formula 

0.8  X  i  X  12 
<M<2=-     — =  2.4. 

If  we  take  bl  =  -^, 
o 


h   —  —  —  W 
*'  ~  6  ~  OT  ' 

By  substituting  />  =  — f- —  in  formulas   (19),   (20), 
0.0025 

(21),  and  (22),  the  following  formulas  may  be  obtained. 
For  arms  having  circular  cross-sections,  of  which  dr  is 
the  diameter, 

(3o). 


TOOTHED   GEARING.  121 

For  elliptical  cross-sections,  a  and  b'  being  the  major 
and  minor  axes  respectively, 


(3i). 

"i 

For  cross-sections,  as  shown  in  Fig.  82, 


Hf  n{ 

For  cross-sections,  as  shown  in  Fig.  83, 


(32)- 


H' 


(33)- 


Example  n. — Taking  the  data  of  Example  10,  re- 
quired the  diameter  for  arms  having  circular  cross- 
sections.  Formula  (30)  gives,  by  substituting  the 
numerical  data, 

=  i.ioSVJ=  i.i5937"=  itt". 


Example  12. — With  the  same  data,  required  the 
dimensions  for  arms  having  elliptical  cross-sections. 
From  formula  (31)  we  have 

I    X    1 2 

tfa2  —  1.356 =  I-356  X  3  =  4.068. 

4 

Assuming  bf  =  \a 

b'a?  =  —  =  4.068 


a  —  \/4^o68~X*2  =s  2" 
fi'=ia=  i". 


122  TOOTHED   GEARING. 

Example  13.  — Using  the  same  data,  it  is  required  to 
determine  the  dimensions  for  flanged  arms  having  cross- 
sections,  such  as  shown  in  Fig.  83. 

From  formula  (33) 

—  b,,h,t      0.8  X  i  X  12 


H' 


=  2.4. 


Let  us  take  B  =  ///,=  £#"'  and    -=  j/f:  hence 


=  h,,  =  iff'  =0.862"  = 


_.    £     A 


and 


More  often  than  otherwise,  the  arms  of  gear-wheels 
are  made  straight,  as  in  Fig.  81  :  sometimes,  however, 
especially  in  large  gears  and  in  gears  subjected  to 
violent  shock  and  strain,  curved  arms  are  preferred,  as 
tending  to  stiffen  and  support  the  rim  better.  Also 
curved  arms,  as  a  general  rule,  cast  better.  When 
single  curved  arms  are  used,  they  may  be  constructed 
as  follows  :  — 

After  having  determined  the  number  of  arms  by  one 
of  the  foregoing  rules,  and  having  marked  their  cen- 
tres A,  C  (Fig.  84),  upon  the  circumference  ABC,  take 
the  arc  AB  =  f  arc  A  C,  and  draw  the  radial  line  OB. 
From  the  centre  O  of  the  wheel,  erect  the  line  OD  per- 
pendicular to  OB,  and  find  upon  OD,  by  trial,  the  centre 


TOOTHED   GEARING. 


123 


a  for  a  circular  arc  passing  through  the  points  O  and  A. 
This  arc  is  the  axis  of  the  arm.  Lay  off,  as  shown  in 
the  figure,  //  (</,  a,  or  //',  according  as  the  cross- 
sections  are  to  be  rectangular,  circular,  elliptical,  or 
flanged*)  at  the  rim,  and  not  less  than  \h 
-£//')  at  the  nave.  Find 
upon  OD,  by  trial,  the 
centres  b  and  c  for  the  Fig.84 
arcs  gk  and  df,  which 
determine  the  form  of 
the  arms. 

Fig.  85  shows  anoth- 
er method  for  drawing 
curved  arms.  Through 
the  centre  o  of  the  wheel 
draw  the  line  oA,  making  30°  with  the  horizontal. 
Draw  also  the  line  AB,  making  60°  with  the  horizon- 
tal. The  point  B  is  the  centre  for  the  axis  oA  of  the 


Fig.85 


arm.  Lay  off,  as  before, 
//  and  |/r,  and  find  upon 
the  line  oB  the  centres 
for  the  arcs  df  and  gk' . 

Double  curved  arms 
are  sometimes  used  for 
large  gears.  Fig.  86 
shows  a  simple  method  ~ 
for  their  construction. 
Draw  the  radial  line  oA,  making  30°  with  the  horizon- 
tal. Take  oc  =  \oA,  and  through  the  point  c  draw 
the  line  pD,  making  60°  with  the  horizontal.  Intersect 

*  The  cross-sections  of  curved  arms  are  generally  elliptical,  the  curved 
form  giving  sufficient  stiffness  to  dispense  with  flanges,  etc. 


124 


TOOTHED   GEARING. 


the  line  pD  by  a  horizontal  line  through  the  point 
A :  the  points  D  and  /  are  respectively  the  centres 
for  the  arcs  oc  and  cA,  which  together  form  the  axis 

Fig. 86 


of  the  arm.  Lay  off  the  arm  widths  as  shown  in  the 
figure.  From  the  point  /  as  a  centre  strike  the  arcs 
ab  and  cf,  and  find  upon  the  line  oD  the  centres  for  the 
remaining  arcs  bd  and//&'. 

Fig. 87 


Another  very  similar  method  for  laying  out  double 
curved  arms  is  shown  in  Fig.  87.  Draw  the  radial  line 
oA,  making  45°  with  the  horizontal.  Take  oc=.^oA, 
and  through  the  point  c  draw  the  vertical  line  pD.  In- 


TOOTHED   GEARING.  125 

tersect  the  line  pD  by  the  horizontal  line  Ap.  The 
points  /  and  D  are  the  centres  for  the  arcs  of  the  axis. 
Lay  off  //  and  |//,  as  shown  in  the  figure,  and  proceed, 
as  in  Fig.  86,  to  strike  the  arcs  ab,  ef,  bd,  and  fkf. 

Rim  :  For  the  thickness  of  the  rim  in  the  plane  of 
the  wheel,  t  (Fig.  87),  Reuleaux  gives  the  formula 

/=0.12-f0.4/  (34) 

in  which  t  is  the  rim  thickness,  and/  the  pitch. 

Rule.  —  To  determine  the  thickness  of  the  rim  of  a 
cast-iron  gear-wheel,  multiply  the  pitch  by  0.4,  and  to 
this  product  add  o.  12". 

Example  14.  —  Required  the  thickness  of  rim  for  a 
gear  having  a  pitch  of  3^".  From  formula  (34) 

/=  0.12  -f  0.4  x  3.5  =  0.12  4-  1.40  =  1.52" =  i|". 

A  simple  and  not  very  accurate  rule  in  use  in  the 
shops  is  to  take  the  rim  thickness  equal  to  f  the  pitch. 

Nave :  The  old  formulas  for  the  thickness  of  the 
nave  —  (k,  Fig.  85)  k—\p  and  k=^d,  in  which  k  is 
the  nave  thickness,/  the  pitch,  and  </the  diameter  of  the 
eye  of  the  wheel  —  are  probably  nearly  correct,  notwith- 
standing their  simplicity.  Unwin  gives  the  formula 

*  =  o.4^  +  i        (35) 

in  which/  is  the  pitch  of  the  teeth,  and  R  the  radius  of 
the  wheel. 

Ride. — To  determine  the  thickness  of  the  nave  of  a 
cast-iron  gear  wheel,  multiply  the  square  of  the  pitch  by 
the  radius  of  the  wheel,  extract  the  cube  root  of  this 
product,  multiply  the  result  by  0.4,  and  add  £". 


126  TOOTHED   GEARING. 

Example  15.  — The  diameter  of  a  gear  is  36"  and  the 
pitch  ij".  Required  the  thickness  of  the  nave.  From 
formula  (35)  we  have 


k  =  o.4'V(i-|)2  X  18  +  0.5  =  0.4/40-5  +  0.5 

=  0.4  x  3.434  +  0.5  =  1.874"  =  i  J". 
By  the  formula  k  —  |/,  we  would  have 

£  =  0.75  x  1.5  =  1.125"=  if- 

Thus,  in  this  case  the  difference  between  the  results  of 
the  two  formulas  is  J". 

For  the  length  of  the  nave  we  may  use  the  formula 

/'='+!     (36) 

in  which  I'  is  the  length  of  the  nave,  /  the  face  width  of 
the  teeth,  and  D  the  diameter  of  the  gear. 

Rule. — To  determine  the  nave  length  of  a  gear, 
divide  the  diameter  of  the  gear  by  30  and  to  the  quo- 
tient add  the  face  width  of  the  teeth. 

Example  16.  — The  diameter  of  a  gear  is  60"  and  the 
face  width  of  the  teeth  8".  Required  the  length  of 
the  nave.  Formula  (36)  gives 

/'=  8  +  |$  =  8  +  2  =  10". 

According  to  Unwin,  the  length  of  the  nave  should 
never  be  less  than  three  times  its  thickness.  He  gives, 
for  the  length,  the  formula  /'  —  /  4-  o.o6R,  which  agrees 
very  nearly  with  formula  (36). 


TOOTHED    GEARING. 


127 


Fig.88 


Shafts  :  When  a  shaft  is  so  supported  by  its  bear- 
ings as  to  be  subjected  to  a  torsional  strain  only,  as  is 
almost  invariably  the  case  in  gear  shafts  (the  bending- 
strain,  due  to  the  weight  of  the  gear  and  the  pressure 
between  the  gears  in  the  direction  of  a  line  joining 
their  centres,  being  ordinarily  slight  enough  to  be  safely 
neglected),  the  calculation  of  the  proper  strength  for 
the  shaft  may  be  made  as  follows  :  — 

In  Fig.  88,  P  represents  the  total  force  tending  to 
twist  the  shaft,  i.e.,  the  total 
force  transmitted  by  the  gear  ; 
R>  the  distance  from  the  cen- 
tre of  the  shaft  to  the  point 
at  which  the  force  acts,  i.e., 
the  radius  of  the  gear ;  and 
d,  the  diameter  of  the  shaft. 
The  greatest  safe  torsional 
strain  which  can  be  sustained 
by  the  shaft  is  given  by  the 
expression 

„  _  *?**  _ 


(ZED 


in   which  f  is    the    greatest    safe    shearing-stress    in 
pounds  per  square  inch  for  the  material  of  the  shaft. 

From  this,  

PR 


°-T9635/ 


or 


, 
d  —  i. 


(37). 


Rule.  — To  determine  the  diameter  of  a  gear  shaft  of 
any  material,  multiply  the  total  force  transmitted  by 


128  TOOTHED   GEARING. 

the  gear  by  the  radius  of  the  gear,  divide  this  product 
by  the  greatest  safe  shearing-stress  in  pounds  per 
square  inch  for  the  material  of  the  shaft,  extract  the 
cube  root  of  the  quotient  thus  obtained,  and  multiply 
the  result  by  1.720. 

Example  1 7.  —  Required  the  diameter  for  an  oak 
shaft,  upon  which  is  a  60"  gear  transmitting  a  force  of 
1,000  pounds,  taking  /' —  500  pounds.  From  formula 
(37), 


=  1.720  x  3-915  =  6.734"  =  6«". 


We  propose  to  take,  for  steel,  /'  =  12,000  pounds; 
for  wrought-iron,  f  =  8,000  pounds  ;  and,  for  cast-iron, 
/'  =  4,000  pounds.  These  values  of  f  are  nearly  mean 
between  those  used  by  Stoney,  Haswell,  and  Unwin, 
which  differ  far  more  than  is  conducive  to  any  degree 
of  accuracy.  Substituting  the  above  values  of  f  suc- 
cessively in  formula  (37),  and  reducing,  we  obtain; 


For  steel,  d  =  v.v\$PR         (38) 

For  wrought-iron,  d  =  o.o86'V/^         (39) 
For  cast-iron,         d  '=  0.108  ^fPR         (40)* 

Rule.  —  To  determine  the  diameter  for  a  gear  shaft 
of  steel,  wrought  or  cast  iron,  multiply  the  total  force 
transmitted  by  the  radius  of  the  gear,  extract  the  cube 
root  of  the  product,  and  multiply  the  result  by  0.075  for 
steel,  0.086  for  wrought-iron,  and  o.  108  for  cast-iron. 

Example  18.  —  A  48"  gear  transmits  a  force  of 
100,000  pounds.  Required  the  diameter  for  a  steel 


TOOTHED    GEARING.  1  2g 

shaft.     From  formula  (38)  we  have 
d=  0.075^100000  x  24  =  0.075  x  62.145  =  4-66v=  4ff". 

Example  19. — Taking  the  data  of  Example  18,  re- 
quired the  diameter  for  a  shaft  of  cast-iron.  Formula 
(40)  gives 


d  =  o.ioSViooooo  x  24  =  0.108  X  62.145  =  6.712"=  6|-f". 

Formulas  for  the  diameters  of  gear  shafts,  in  terms  of 
the  horse-power  transmitted  and  the  revolutions  per 
minute,  may  be  obtained  as  follows  :  — 

As  before  explained,  we  have  the  expression 

^=63000^ 

H  representing  the  horse-power,  R  the  radius  of  the 
gear,  and  ;/  the  number  of  revolutions  per  minute. 
Substituting  this  value  of  P  in  formulas  (37),  (38),  (39), 
and  (40),  and  reducing,  we  obtain  the  following  :  — 

fjr 

General  formula,     d—  68.44  y-y/        (41) 


.984^-^ 


For  steel,  </=    2.984-  (42) 


For  wrought-iron,    d—    3.422^—          (43) 


js 

=    4.297V" 


For  cast-iron,  d=    4.297V-  (44)« 


I3O  TOOTHED   GEARING. 

Rule.  —  To  determine  the  diameter  for  a  gear  shaft 
of  any  material,  from  the  horse-power  and  number  of 
revolutions  per  minute,  divide  the  horse-power  by  the 
product  of  the  number  of  revolutions  into  the  greatest 
safe  shearing-stress  in  pounds  per  square  inch  for  the 
material  of  the  shaft,  extract  the  cube  root  of  the  quo- 
tient thus  obtained,  and  multiply  the  result  by  68.44. 

To  determine  the  diameter  for  a  gear  shaft  of  steel, 
wrought  or  cast  iron,  from  the  horse-power  and  number 
of  revolutions  per  minute,  divide  the  horse-power  by 
the  number  of  revolutions,  extract  the  cube  root  of  the 
quotient,  and  multiply  the  result  by  2.984  for  steel, 
3.422  for  wrought-iron,  and  4.297  for  cast-iron. 

Example  20.  —  Required  the  diameter  for  an  oak 
gear  shaft  which  transmits  a  force  of  zo-horse  power, 
and  makes  40  revolutions  per  minute.  If  we  take  for 
the  greatest  safe  shearing-stress  for  oak  f  =  500  pounds 
per  square  inch,  we  shall  have,  from  formula  (41), 

d  =  68.44V5/ — =  68.44V3/— ^—  =  68.44  X  — 4r 

"V  4°  *  500  V  2000  1 2.60 

=  5432"  =  5*"  nearly. 

Example  21. — Taking  the  data  of  Example  20,  re- 
quired the  diameters  for  shafts  of  steel  and  wrought- 
iron.  From  formula  (42), 

</=  2.984^  =  2.984^0^5"  =  2.984x0.62996=  1.88"  =  iff" 
for  steel.     From  formula  (43), 

d=  3422\/H  =  3-422  x  0.62996  =  2.1557"=  2-gs" 
for  wrought-iron. 


TOOTHED   GEARING.  131 

Convenient  formulas  for  gear-shaft  diameters  in  terms 
of  the  pitch  and  radius,  may  be  obtained  in  the  following 
manner.  From  formula  (12,  b)  we  have,  as  before, 


0.0025 


=  400/2 


which  value,  substituted  in  formulas  (37),  (38),  (39),  and 
(40),  gives  the  following  formulas  :  — 

General  formula,     ^=12.673^/^-77-         (45) 

For  steel,  d=    o.  $$$#*£  (46) 

For  wrought-iron,    <t  =    o.634V//2^  (47) 

For  cast-iron,          d—    0.796^^  (48). 

Rule.  —  To  determine  the  diameter  of  a  gear  shaft  of 
any  material,  from  the  pitch  and  radius  of  the  gear,  mul- 
tiply the  square  of  the  pitch  by  the  radius,  divide  the 
product  by  the  greatest  safe  shearing-stress  in  pounds 
per  square  inch  for  the  material  of  the  shaft,  extract 
the  cube  root  of  the  quotient  thus  obtained,  and  mul- 
tiply the  result  by  12.673.  To  determine  the  diameter 
of  a  gear  shaft  of  steel,  wrought  or  cast  iron,  from  the 
pitch  and  radius  of  the  gear,  multiply  the  square  of 
the  pitch  by  the  radius,  extract  the  cube  root  of  the 
product,  and  multiply  the  result  by  0.553  for  steel,  0.634 
for  wrought-iron,  and  0.796  for  cast-iron.* 

*  The  expression  P  =  4oo/2  is  true  only  for  cast-iron  gears  :  hence  the 
value  of  pz  in  formulas  (45),  (46),  (47),  and  (48),  must  be  for  a  cast-iron 
gear. 


132  TOOTHED   GEARING. 

Example  22.  —  A  cast-iron  gear  has  a  diameter  ot 
12"  and  a  pitch  of  J'1  '.  Required  the  diameter  for  a 
brass  shaft,  supposing  f  =  3,000  pounds  for  brass. 
From  formula  (45) 


=  i2.673'</o.ooo5  =  12.673  X  0.07937=  i 


Example  23.  —  The  diameter  of  a  cast-iron  gear  is 
60"  and  the  pitch  2".  Required  the  diameters  for 
shafts  of  steel  and  wrought-iron.  From  formula  (46) 


<^=  °-553'V4  X  30  =  o.553Vi20  =  0.553  X  4-932 

=  2.727"  =2jr 

for  steel.     From  formula  (47)  we  have 

</=  0.634^4  X  30  =  0.634  x  4-932  =  3-«  7"  =  3t" 

for  wrought-iron. 

Gear  shafts  are  most  commonly  of  wrought-iron  : 
when,  however,  wrought-iron  shafts,  in  order  to  give 
the  necessary  strength,  become  so  large  as  to  be  in- 
convenient, steel  shafts  are  used.  Cast-iron  shafts  are, 
as  a  rule,  unreliable  and  treacherous  ;  they  are  there- 
fore seldom  used,  except  for  the  transmission  of  slight 
powers,  and  in  cheap,  inferior  machinery.  The  follow- 
ing tables,  calculated  from  formulas  (38),  (39),  (42),  and 
(43),  to  the  nearest  ^",  will  be  found  very  convenient 
in  designing  gear  shafts  of  steel  and  wrought-iron  :  — 


TOOTHED  GEARING, 


133 


TABLE  VI. 


PR 

//for 
steel. 

//for 
wrought 
iron. 

PR 

//for 
steel. 

//for 
wrought 
iron. 

250 

H" 

ft" 

60000 

2H" 

3ir 

500 

ti 

ti 

70000 

3& 

3H 

IOOO 

! 

ft 

80000 

3H 

3tt 

1500 

if 

H 

90000 

3ft 

3tt 

2000 

IT 

T/T 

IOOOOO 

3H 

4 

2500 

*A 

«tt 

1  1  0000 

3li 

4i 

3000 

'& 

'54 

120000 

3« 

4H 

3500 

*& 

Jft 

130000 

3ff 

41! 

4000 

lT36 

rl! 

140000 

3il 

4H 

4500 

IB! 

I§J 

1  50000 

3f! 

4ft 

5000 

1^ 

1  1| 

175000 

4ef 

411 

6000 

if! 

i  A 

200000 

4|f 

5^ 

7000 

i* 

250000 

4M 

sH 

8000 

'1 

iff 

500000 

sH 

^f! 

IOOOO 

Iff 

iff 

750000 

6if 

7ii 

12500 

'! 

2 

IOOOOOO 

7! 

8fl 

15000 

\\\ 

2i 

1  500000 

^11 

9§J 

20000 

2^2 

2§i 

2000000 

91! 

io|| 

25000 

2ft 

2ff 

2500000 

n|! 

30000 

* 

2e! 

3000000 

ioif 

I2M 

35000 

2if 

3500000 

nit 

!3ft 

40000 

2ft 

4f 

4000000 

"H 

r3ii 

45000 

Hf 

3ft 

4500000 

I2| 

I4ii 

50000 

Hi 

3H 

5000000 

I2|f 

•4H 

134 


TOOTHED  GEARING. 


TABLE   VII. 


H 

n 

dTfor 
steel. 

dTfor 

wrought 
iron. 

H 

n 

steel. 

^for 
wrought 
iron. 

0.025 

i" 

i  " 

3-75 

411" 

5T5e" 

0.050 

'& 

l\\ 

4 

4ef 

sA 

0.075 

'H 

"A 

4.25 

4e4 

5*4 

O.I  00 

1  IT 

TM 

4.50 

4l6 

sft 

0.150 

'*i 

i  ^^ 

4-75 

sA 

54 

O.2OO 

if 

2 

5 

sA 

511 

0.250 

\\ 

~3£ 

5.50 

sH 

6A 

0.300 

2 

2el 

6 

5*4 

6/5 

0.350 

2A 

•"3"2 

6.50 

5i96 

6|| 

0.400 

2i4 

2§| 

7 

514 

^6T 

O.5OO 

2| 

2f| 

8 

5fi 

6fJ 

0.600 

2f| 

2if 

9 

6^j 

71 

0.700 

23'2 

3  A 

10 

52.7 

71 

0.800 

2|| 

3H 

ii 

6|| 

7*4 

O.pOO 

26T 

3A 

12 

6^T 

'  7*4 

I 

2|| 

3il 

14 

7l6" 

ST 

1.25 

3A 

3ii 

16 

7*4 

8| 

1.50 

364 

3  1| 

18 

7M 

8M 

i-75 

3H 

4i 

20 

8A 

93% 

2 

311 

4^ 

22 

8ff 

911 

2.25 

3f| 

4*1 

25 

m 

10 

2.50 

4A 

4H 

27 

8fi 

IOH 

2.75 

4H 

4eT 

30 

9^T 

i  of 

3 

4A 

41! 

32 

9M 

io|| 

3-25 

4|| 

sA 

35 

9*4 

"A 

3-50 

4^ 

5A 

40 

I°tt 

»*4 

TOOTHED   GEARING.  135 

Example  24.  —  Required  the  diameter  for  a  wrought- 
iron  shaft  for  a  40"  gear  which  transmits  a  force  of 
10,000  pounds.  In  this  case 

PR  =   IOOOO  X    2O  =  2OOOOO 

and,  from  Table  VI.,  the  value  of  d  for  wrought-iron 
corresponding  to  PR  =  200,000  is  d  =  S^"- 

Example  25.  —  The  diameter  of  a  wrought-iron  gear 
shaft  is  4^".  Required  the  force  which  the  shaft  can 
safely  transmit  by  means  of  a  24"  gear.  From  Table 
VI.  the  value  of  PR  corresponding  to  d  —  4^"  for 
wrought-iron  is  1 10,000  :  hence  we  will  have 

I IOOOO          I IOOOO 

p  =  — — —  = =  9167  pounds  nearly. 

Example  26.  — A  gear  transmitting  a  force  of  2O-horse 
power  makes  200  revolutions  per  minute.  Required  the 
diameter  for  a  shaft  of  steel.  We  have 

H          20  I 

—  = =  —  =  o.ioo 

n       200       10 

and,  from  Table  VII.,  the  value  of  d  for  steel  corre- 

TT 

spending  to  —  =o.  100  is  d  =  I  |f  ". 

Example  27.  —  A  2"  steel  shaft  transmits  a  force  of 
2 5 -horse  power.  It  is  required  to  determine  the  proper 
number  of  revolutions  per  minute.  From  Table  VII. 

TT 

the  value  of  —  which  corresponds  to  d  =  2!'  for  steel  is 
;/ 

TT 

—  =  0.300 :  hence  we  have 
« 

H     25 

—  =  —  =  0.300 
n        n 

or 

n  =  83^  revolutions  per  minutt. 


136  TOOTHED   GEARING. 

Keys  :  We  may  take  for  the  mean  width  of  the  key 
which  fixes  the  gear  upon  its  shaft  »S  =  o.2&/,  and,  for 
the  thickness,  S'  =  0.014^;  vS  and  S'  being  respectively 
the  mean  width  and  thickness  of  the  key,  and  d  the 
diameter  of  the  shaft.  More  accurately,  according  to 
Reuleaux, 

S=  o.i6  +  |         (49) 
and 

S'  =  o.i6  +  ^         (50). 

Rule.  —  To  determine  the  mean  width  of  the  fixing- 
key,  divide  the  diameter  of  the  shaft  by  5,  and  to  the 
result  add  o.  16".  To  determine  the  key  thickness,  di- 
vide the  diameter  of  the  shaft  by  10,  and  to  the  result 
add  o.i  6". 

Example  28.  —  Required  the  mean  width  and  thick- 
ness for  a  fixing-key  of  sufficient  strength  for  the  gear 
and  shaft  given  in  Example  24.  From  formula  (49)  we 
have 

S=  0.16  +  5'°3125  =  0.16  +  1.00625  =  1.16625"  =  itt"- 
From  formula  (50), 
S'=  0.16  +  5'°3125  =  0.16  +  0.503125  =  0.663125"=  |f". 

Weight  of  Gears :  The  approximate  weight  of  a 
spur-wheel  may  be  calculated  by  the  following  formula, 
given  by  Reuleaux.  G  represents  the  approximate 
weight  in  pounds,  N  the  number  of  teeth,  and  /  and  / 
respectively  the  pitch  and  face  width  :  — 

0.0014^*)         (51). 


TOOTHED   GEARING.  137. 

Rule. — To  determine  the  approximate  weight  of  a 
spur  wheel,  add  0.215  times  the  number  of  teeth  to 
0.0014  times  the  square  of  the  number  of  teeth,  and 
multiply  the  sum  by  the  product  of  the  square  of  the 
pitch  into  the  face  width. 

Example  29.  —  Required  the  approximate  weight  of  a 
spur  wheel  having  50  teeth,  a  pitch  of  2",  and  a  face 
width  of  4j".  From  formula  (51)  we  have 

G  =  4j  x  22(o.2i5  x  50  -f-  0.0014  x  so2) 

=  18(10.75  +  3-5°)  =  18  x  14-25  =  256.50  pounds. 

When  the  face  width  is  twice  the  pitch  (/==2/),  for- 
mula (51)  becomes 

G  =  2/3  (0.2 1 57V  +  O.OOI47V2) 
or 

G  =/3(o.430^+  0.0028^)         (52). 

Rule.  —  To  determine  the  approximate  weight  of  a 
spur  wheel  when  the  face  width  is  equal  to  twice  the 
pitch,  add  0.430  times  the  number  of  teeth  to  0.0028 
times  the  square  of  the  number  of  teeth,  and  multiply 
the  sum  by  the  cube  of  the  pitch. 

Example  30.  —  Required  the  approximate  weight  of 
a  spur  wheel  having  50  teeth,  a  pitch  of  2",  and  a  face 
width  of  4".  From  formula  (52)  we  have 

G  =  2  (0.430  X  50  +  0.0028  X  5Q2)  =  8  X  28.50=  228  pounds. 

The   following   table,   computed    from   formula  (51), 

/"• 
gives  values  of  —  for  different  numbers  of  teeth  :  — 


138 


TOOTHED   GEARING. 
TABLE    VIII. 


N 

o 

2 

4 

6 

8 

20 

4.86 

541 

5-97 

6.54 

7.12 

30 

7.71 

8.3  r 

8-93 

9-55 

10.19 

40 

10.84 

11.50 

12.17 

12.85 

13-55 

50 

14.25 

14.97 

15.69 

16.43 

17.18 

60 

17.94 

18.71 

19.49 

20.29 

21.09 

70 

21.91 

22.74 

23.58 

2443 

25.29 

80 

26.16 

27.04 

27.94 

28.84 

29.76 

90 

30.69 

31-63 

32.58 

33-54 

34.52 

100 

35-50 

36.50 

37-50 

38.52 

39-55 

120 

45.96 

47.07 

48.19 

49-32 

50.46 

140 

57-54 

58.76 

59-99 

61.23 

62.49 

160 

70.24 

71-57 

72.91 

74.27 

75-63 

1  80 

84.06 

85.50 

86.96 

88.42 

89.90 

2OO 

99.00 

100.56 

102.12 

103.70 

105.29 

220 

115.06 

116.73 

Il8.4I 

120.10 

121.80 

Example  31. — Required  the  approximate  weight  of  a 
spur  wheel  having  126  teeth,  the  pitch  being  3"  and  the 

face   width    f.     From   Table  VIII.   the  value  of   ~, 
which  corresponds  to  N=  126,  is  49.32  :  hence 

G 

7X  32~ 
(7=49.32  X  7  X  9  =  3107.16  pounds. 

To  determine  the  approximate  weight  of  a  bevel  gear, 
proceed  as  explained  in  the  above  example  for  a  spur 
wheel,  except  that  the  .  tabular  number  must  be 
multiplied  by  0.855. 


TOOTHED    GEARING.  139 

Example  32.  —  Required  to  determine  the  approxi- 
mate weight  of  a  bevel  wheel,  for  which  N  —  48, 

/* 
p  —  3",  and  /  =  7".      From  the  table  the  value  of  ~ 

corresponding  to  ^¥=48  is  13.55.     This  multiplied  by 
0.855  gives  1 1.585  :  hence 

—  =11.585 

7  X9 

G—  11.585  x  7  X  9  =  729.855  pounds. 

§  XV.  —  Recapitulation  of  Formulas  and  Rules. 

For  convenience  in  designing,  the  various  rules  and 
formulas  developed  in  the  foregoing  pages  have  been 
gathered  together  in  the  following  recapitulation  :  — 

NOTATION. 

R    =  radius  of  the  pitch  circle. 
D  =  diameter  of  the  pitch  circle. 
C    =  circumference  of  the  pitch  circle. 
TT     =  constant  3.14159. 
/    =  circumferential  pitch. 
pd  —  diametral  pitch. 
N  —  number  of  teeth. 
I"   =  length  of  chord  subtending  the  pitch. 
n     =  number  of  revolutions  per  minute. 
P    =  total  force  transmitted. 
W  =  total  force  transmitted. 
v     =  circumferential  velocity  in  feet  per  second. 
V    =  circumferential  velocity  in  feet  per  second. 
/    =  greatest  safe  working-stress  in  pounds  per  square  inch 

for  the  material. 
/     =  face  width. 


140  TOOTHED   GEARING. 

h     =  total  height  of  teeth. 

h'    —  height  of  teeth  below  pitch  circle. 

h"  =  height  of  teeth  above  pitch  circle. 

b     =  breadth  of  teeth  at  pitch  circle. 

H  =  horse-power  transmitted. 

72 /  =  number  of  arms. 

Jit    =  width  of  rectangular  arms  in  plane  of  the  pitch  circle. 

bl    =  thickness  of  rectangular  arms. 

x    —  variable  co-efficient. 

d'   —  diameter  of  circular  arms. 

//',  B,  h,,,  and  b,,  =  dimensions  for  Figs.  82  and  83. 

a     =  major  axis  for  elliptical  arms. 

b'    =  minor  axis  for  elliptical  arms. 

/     =  thickness  of  rim. 

k     =  thickness  of  nave. 

/'    =  length  of  nave. 

d    —  diameter  of  shaft. 

S    =  mean  width  of  fixing-key. 

S'  =  thickness  of  fixing-key. 

G  =  approximate  weight  of  spur  wheel. 

Dimensions  are  in  inches,  forces  and  weights  in  pounds, 
unless  otherwise  stated. 

C=TrD=2irR  (l). 

Rule.  — To  find  the  circumference  of  the  pitch  circle, 
multiply  the  diameter  by  3.14159,  or  the  radius  by 
6.28318. 

Z>=-,    R~-        (2). 

7T  27T 

Rule. — To  find  the  diameter  of  the  pitch  circle,  di- 
vide the  circumference  by  3.14159.  To  find  the  radius, 
divide  the  circumference  by  6.28318. 


TOOTHED   GEARING.  141 

C  C 

W=—,     C=Nj>,    P—^T        (3). 

Rule.  —  To  find  the  number  of  teeth,  divide  the  cir- 
cumference by  the  pitch.  To  find  the  circumference, 
multiply  the  number  of  teeth  by  the  pitch.  To  find 
the  pitch,  divide  the  circumference  by  the  number  of 
teeth. 

_N_ir_          _  jr_ 

Rule.  —  To  find  the  diametral  pitch,  divide  the  num- 
ber of  teeth  by  the  diameter,  or  divide  3.14159  by  the 
pitch.  To  find  the  pitch,  divide  3.14159  by  the  diame- 
tral pitch. 

Rule.  —  To  find  the  length  of  the  chord  which  sub- 
tends the  pitch,  multiply  twice  the  radius  by  the  natural 
sine  of  half  the  angle  limited  by  the  pitch. 

(6). 

Rule.  —  To  find  the  length  of  the  chord  which  sub- 
tends the  pitch,  divide  180°  by  the  number  of  teeth, 
take  the  natural  sine  of  the  angle  thus  obtained,  and 
multiply  by  the  diameter. 

'n'~~N~~R~T>~~~C         '"" 

Rule.  —  The  ratio  of  the  numbers  of  revolutions  of  a 
pair  of  gears  is  inversely  proportional  to  the  ratio  of 
their  numbers  of  teeth  to  the  ratio  of  their  radii, 
diameters,  or  circumferences. 


142 


TOOTHED    GEARING, 


=-r         (8). 

Rule.  —  The  ratio  of  the  powers  of  two  gears  on  the 
same  shaft  is  inversely  proportional  to  the  ratio  of  their 
radii. 


(9). 


W      V  PV  Wv 

-—  =  —,     W=  —  )    P=  -jy 
P       v  v  V 

R^ile.  —  The  ratio  of  the  powers  of  two  gears  on  the 
same  shaft  is  inversely  proportional  to  the  ratio  of  their 
circumferential  velocities. 


(10). 


Rule.  — To  find  the  pitch  for  a  gear  of  any  material, 
divide  the  force  transmitted  by  the  greatest  safe  work- 
ing-stress in  pounds  per  square  inch  for  the  material, 
multiply  the  quotient  by  the  ratio  of  the  pitch  to  the 
face  width,  extract  the  square  root  of  the  product,  and 
multiply  the  result  by  4.93. 

For  cast-iron.* 


Violent  shock,  /  =  o.oySy/  P  x  ~  (a) 
Moderate  shock,  /  =  0.07  y  P  X  ^  (b) 
Little  or  no  shock,  /  =  0.05  y  P  x  ~  (c) 


(ii). 


Rule. — To  find  the  pitch  for  a  cast-iron  gear,  multi- 
ply the  force  transmitted  by  the  ratio  of  the  pitch  to  the 
face  width,  extract  the  square  root  of  the  product,  and 


*  h  =  o.;/,  ti  —  o-4/,  h"  —  0.3^,  and  b 


TOO  THED   GEA AYA'C. 


143 


multiply  the  result  by  0.078  for  violent  shock,  0.07  for 
moderate  shock,  or  0.05  for  little  or  no  shock. 

When  /  =  2p, 

Violent  shock,         /  =  0.055^  (a) 

Moderate  shock,      /  =  0.05  ^P  (b)  (12). 

Little  or  no  shock,  /  =  0.035^  (c) 

Rule. — To  find  the  pitch  for  a  cast-iron  gear  when 
the  face  width  is  twice  the  pitch,  multiply  the  square 
root  of  the  force  transmitted  by  0.055  for  violent  shock, 
0.05  for  moderate  shock,  or  0.35  for  little  or  no  shock. 


Violent  shock,        /  =  1.83! 

f 

/  J-f          *h 

Moderate  shock,     /  =  i.64y  —  x  -,     (b} 


Little  or  no  shock 


c,/=i.i7\/f  x^ 


7       (<) 


(13) 


Rule.  —  To  find  the  pitch  for  a  cast-iron  gear  from 
the  horse-power  transmitted  and  circumferential  velocity 
in  feet  per  second,  divide  the  horse-power  by  the  cir- 
cumferential velocity,  multiply  the  quotient  by  the  ratio 
of  the  pitch  to  the  face  width,  extract  the  square  root  of 
the  product,  and  multiply  the  result  by  1.83  for  violent 
shock,  1.64  for  moderate  shock,  or  1.17  for  little  or  no 

shock. 

When  /  =  2/, 

I  H 
Violent  shock,          /=i.29y  —     (a) 


Moderate  shock,      /  = 
Little  or  no  shock,  /  = 


f- 


144 


TOOTHED   GEARING. 


Rule.  —  To  find  the  pitch  for  a  cast-iron  gear,  from 
the  horse-power  and  velocity,  when  the  face  width  is 
twice  the  pitch,  divide  the  horse-power  by  the  velocity, 
extract  the  square  root  of  the  quotient,  and  multiply 
the  result  by  1.29  for  violent  shock,  1.17  for  moderate 
shock,  or  0.82  for  little  or  no  shock. 


Violent  shock,        /  =  sy.yiy  -=-  x  j     (a) 


Moderate  shock,     p  =  24.84^7  -       x        (V) 


Little  or  no  shock,  /  =  I7-72y  ^  x  /      (c) 


(15) 


Rule.  —  To  find  the  pitch  for  a  cast-iron  gear  from 
the  horse-power  and  number  of  revolutions  per  minute, 
divide  the  horse-power  by  the  product  of  the  diameter 
into  the  number  of  revolutions,  multiply  the  quotient 
by  the  ratio  of  the  pitch  to  the  face  width,  extract  the 
square  root  of  the  product,  and  multiply  the  result  by 

27.71  for  violent  shock,  24.84  for  moderate  shock,  or 

17.72  for  little  or  no  shock. 


When  /  =  2/, 
Violent  shock,         /  «=  19. 


•72\/ 


Moderate  shock,      /=  17 


Little  or  no  shock,  p  =  12.42^  — 


H_ 
Dn 
Jf 


(16). 


Rule.  —  To  find  the  pitch  for  a  cast-iron  gear,  from 
the  horse-power  and  number  of  revolutions  per  minute, 
when  the  face  width  is  twice  the  pitch,  divide  the  horse- 


TOOTHED    GEARING.  145 

power  by  the  product  of  the  diameter  into  the  number 
of  revolutions,  extract  the  square  root  of  the  quotient, 
and  multiply  the  result  by  19.54  for  violent  shock,  17.72 
for  moderate  shock,  or  12.42  for  little  or  no  shock. 

/,i;,Ia  =  -^-7         (17). 
5oo;// 

Rule.  —  To  find  the  quantity^//,2  (the  thickness  of 
the  arm  multiplied  by  the  square  of  the  width)  for  cast- 
iron  arms,  multiply  the  force  transmitted  by  the  radius 
of  the  pitch  circle,  and  divide  the  product  by  500  times 
the  number  of  arms. 


(18). 

Rule.  —  To  find  the  width  of  the  arms  in  the  plane 
of  the  pitch  circle,  multiply  the  force  transmitted  by 
the  radius  of  the  pitch  circle,  extract  the  cube  root  of 
the  product,  and  multiply  the  result  by  the  tabular 
number  (in  Table  V.)  corresponding  to  the  required 

number  of  arms  and  value  of  -i. 


Rule.  —  To  find  the  diameter  for  cast-iron  arms 
having  circular  cross-sections,  multiply  the  force  trans- 
mitted by  the  radius  of  the  pitch  circle,  divide  the 
product  by  the  number  of  arms,  extract  the  cube  root 
of  the  quotient,  and  multiply  the  result  by  0.15. 

PR 

b'a?  =  0.00339  —7         (20). 
nl 

.  —  To  find  the  quantity  of  b'a2  (the  minor  axis 


146  TOOTHED   GEARING. 

of  elliptical  cross-section  multiplied  by  the  square  of  the 
major  axis)  for  cast-iron  arms,  multiply  the  force  trans- 
mitted by  the  radius  of  the  pitch  circle,  divide  the 
product  by  the  number  of  arms,  and  multiply  the  result 
by  0.00339. 

b,,H's  +  Bh,t        PR 

(21)  * 


H'  ~  500;;, 

-  b,,h,t        PR 


soon 


(22)    f 


(23). 

Rule.  —  To  find  the  number  of  arms,  extract  the 
fourth  root  of  the  pitch  and  the  square  root  of  the  num- 
ber of  teeth,  multiply  the  two  roots  together,  and  the 
product  by  0.56. 


(24). 

Rule.  —  To  find  the  quantity  bji?  (see  formula  17) 
for  cast-iron  arms,  from  the  horse-power  and  number  of 
revolutions  per  minute,  multiply  the  horse-power  by 
1  26,  and  divide  by  the  product  of  the  number  of  revolu- 
tions into  the  number  of  arms. 


Rule.  —  To  find  the  diameter  for  cast-iron  arms 
having  circular  cross-sections,  from  the  horse-power  and 
number  of  revolutions  per  minute,  divide  the  horse- 
power by  the  product  of  the  number  of  revolutions  into 

*  See  Fig.  82.  t  See  Fig.  83. 


TOOTHED   GEARING.  147 

the  number  of  arms,  extract  the  cube  root  of  the  quo- 
tient, and  multiply  the  result  by  5.969. 

TT 

t'a*  =  2iw—j         (26). 

Rule.  —  To  find  the  quantity  bfa2  (see  formula  20) 
for  cast-iron  arms,  from  the  horse-power  and  number  of 
revolutions  per  minute,  divide  the  horse-power  by  the 
product  of  the  number  of  revolutions  into  the  number 
of  arms,  and  multiply  the  quotient  by  213.57. 


H'  '    nnT 

-b,,h,t       I26H 


. 
^A2  =  -^r-  (29). 

Ride.  —  To  find  the  quantity  bji?  (see  formula  17) 
for  cast-iron  arms,  from  the  pitch,  multiply  o.S  times 
the  square  of  the  pitch  by  the  radius  of  the  pitch  circle, 
and  divide  the  product  by  the  number  of  arms. 

(30). 

Rule.  —  To  find  the  diameter  of  cast-iron  arms  having 
circular  cross-sections,  from  the  pitch,  multiply  the 
square  of  the  pitch  by  the  radius  of  the  pitch  circle, 
divide  the  product  by  the  number  of  arms,  extract  the 
cube  root  of  the  quotient,  and  multiply  the  result  by 
1.105. 

*  See  Fig.  82.  t  See  Fig.  83. 


148  TOOTHED   GEARING, 

b'a*  =  1.356^         (3')- 

"I 

Rule.  —  To  find  the  quantity  b'a2  (see  formula  20) 
from  the  pitch,  multiply  the  square  of  the  pitch  by  the 
radius  of  the  pitch  circle,  divide  the  product  by  the  num- 
ber of  arms,  and  multiply  the  result  by  1.356. 


(3*) 


(33)  t 


H' 


H' 


/  =  o.i2+o.4/  (34). 

Rule.  —To  find  the  thickness  of  the  rim,  add  0.12"  to 
0.4  times  the  pitch. 

*  =  o.4V/^  +  t        (35). 

Rule.  —  To  find  the  thickness  of  the  nave,  multiply 
the  square  of  the  pitch  by  the  radius  of  the  pitch  circle, 
extract  the  cube  root  of  the  product,  multiply  the  root 
by  0.4,  and  to  the  result  add  J". 


Rule.  —  To  find  the  length  of  the  nave,  divide  the 
diameter  of  the  pitch  circle  by  30,  and  to  the  result  add 
the  face  width  of  the  teeth. 


(37). 
Rule. — To  find  the  diameter  of  a  gear  shaft  of  any 

*  See  Fig.  82.  t  See  Fig.  83. 


TOOTHED   GEARING.  149 

material,  multiply  the  force  transmitted  by  the  radius 
of  the  pitch  circle,  divide  the  product  by  the  greatest 
safe  shearing-stress  in  pounds  per  square  inch  for  the 
material,  extract  the  cube  root  of  the  quotient,  and  mul- 
tiply the  result  by  1.720. 

For  steel,  d  =  0.0751^?         (38) 

For  wrought-iron,  d  —  0.086'^^         (39) 

For  cast-iron,  d —  o.io83V '  PR  (40). 
Ride.  —  To  find  the  diameter  of  a  gear  shaft,  multiply 
the  force  transmitted  by  the  radius  of  the  pitch  circle, 
extract  the  cube  root  of  the  product,  and  multiply  the 
result  by  0.075  f°r  steel,  0.086  for  wrought-iron,  and 
0.108  for  cast-iron. 


=  68.44^1 


,     x 
(40. 


Rule. — To  find  the  diameter  of  a  gear  shaft  of  any 
material  from  the  horse-power  and  number  of  revolu- 
tions, divide  the  horse-power  by  the  product  of  the 
number  of  revolutions  into  the  greatest  safe  shearing- 
stress  in  pounds  per  square  inch  for  the  material,  ex- 
tract the  cube  root  of  the  quotient,  and  multiply  the 
result  by  68.44. 

fff 

For  steel,  ^=2.984^—         (42) 

For  wrought-iron,  d  =  3.422^-—         (43) 

3/5" 
For  cast-iron,         ^=4.297^—         (44)« 


150  TOOTHED   GEARING. 

Rule.  —  To  find  the  diameter  of  a  gear  shaft  from  the 
horse-power  and  number  of  revolutions,  divide  the  horse- 
power by  the  number  of  revolutions,  extract  the  cube 
root  of  the  quotient,  and  multiply  the  result  by  2.984 
for  steel,  3.422  for  wrought-iron,  and  4.297  for  cast-iron. 


(45). 

Ride.  —  To  find  the  diameter  of  a  gear  shaft  of  any 
material  from  the  pitch,  multiply  the  square  of  the  pitch 
by  the  radius  of  the  pitch  circle,  divide  the  product  by 
the  greatest  safe  shearing-stress  in  pounds  per  square 
inch  for  the  material  used,  extract  the  cube  root  of  the 
quotient,  and  multiply  the  result  by  12.673. 


For  steel,  d  =  o.P2R         (46) 

For  wrought-iron,   d=o.6i)$p2R         (47) 


For  cast-iron,          d—Q.^lp^R         (48) 

Rule.  —  To  find  the  diameter  of  a  gear  shaft  from  the 
pitch,  multiply  the  square  of  the  pitch  by  the  radius  of 
the  pitch  circle,  extract  the  cube  root  of  the  product, 
and  multiply  the  result  by  0.553  for  steel,  0.634  for 
wrought-iron,  and  0.796  for  cast-iron. 


(49) 


S'=o.i6  +  —          (50). 

Rule.  —  To    find    the   mean    width    of  a   fixing-key, 
divide  the  diameter  of  the  shaft  by  5,  and  to  the  result 


TOOTHED    GEARING.  151 

add  o.  16".     To  find  the  thickness  of  the  key,  divide  the 
diameter  of  the  shaft  by  ro,  and  to  the  result  add  0.16". 


•O.OOI4/V72)          (si). 

Rule. — To  find  the  approximate  weight  of  a  spin- 
wheel,  add  0.215  times  the  number  of  teeth  to  0.0014, 
the  square  of  the  number  of  teeth,  and  multiply  the 
sum  by  the  product  of  the  face  width  into  the  square 
of  the  pitch. 

When  /  =  2p, 
G=p*  (0.4307V  -f  0.00287V2)         (5 2) . 

Rule. — To  find  the  approximate  weight  of  a  spur 
wheel  when  the  face  width  is  twice  the  pitch,  add  0.430 
times  the  number  of  teeth  to  0.0028  times  the  square 
of  the  number  of  teeth,  and  multiply  the  sum  by  the 
cube  of  the  pitch. 

§  XVI.  —  Complete  Design  of  Spur-Wheel,  Bevels,  Worm,  Screw 
Gear,  etc. 

Example  I.  —  Required  to  design  and  make  full  work- 
ing drawings  for  a  36"  cast-iron  spur  wheel  to  transmit 
a  force  of  5,000  pounds,  violent  shock. 

For  the  pitch  we  have,  from  formula  (12,  a), 

p  =  o.o55\/5ooo  =  0.055  X  70-71  =  3-889" 
for  the  face  width, 

/=  2p=  2  x  3.889=  7.778". 
As  explained  in  §  XIII. ,  we  have  for  the  total  height 


152  TOOTHED   GEARING. 

of  the  teeth,  and  heights  below  and  above  the  pitch 
circle, 

h  =  h'  +  h"  =  o.4/  -f  o.3/  =  0.7  x  3.889  =  2.7223" 
^'=0.4  x  3-889=  1.5556" 
^"=0.3  x  3.889=  1.1667". 

We  may  take,  for  the  breadth  of  the  teeth  on  the  pitch 
circle,  b  =  o.^p=  i. 75".  From  formulas  (i)  and  (3), 
for  the  circumference  and  number  of  teeth, 

(7=3.14159  x  36  =  113.10 
and 

113.10  _ 

'IW 
From  formula  (23),  the  number  of  arms  is 

«/=  0.56^29  ^3.889  =  0.56  x  5.385  x  1.40  =  4. 

If  we  wish  to  have  elliptical  cross-sections  for  the  arms, 
we  have,  from  formula  (20), 

,,  qooo  x  1 8 

b  a*  =  0.00339  x =  0.00339  X  22500  =  76.275. 

4 

Taking 

*>'="->    ^2  =  ^=  76.275; 

or,  for  the  major  axis  of  the  cross-section, 


a  =  ^152.55  =  5.343" 
and,  for  the  minor  axis, 

£'=^=2.6715". 


TOOTHED   GEARING.  153 

For  the  thickness  of  the  rim,  from  formula  (34), 
/=  0.12  -f  0.4  x  3.889  =  0.12  4-  1.5556  =  1.6756." 
Formula  (35)  gives,  for  the  thickness  of  the  nave, 

k  =  0.4V3-8892  x  18  4-  \  =  0.4  x  6.481  -f  \  =  3.092". 
The  length  of  the  nave  is,  from  formula  (36), 

/'  =   7.778  -f  f§-  =   7.778  4-  1.2  =  8.978". 

Formula  (39)  gives,  for  the  diameter  of  the  wrought-iron 
shaft, 


d—  0.086^5000  x  18  =  0.086  x  44.814  =  3.854". 

For  the  mean  width  and  thickness  of  the  fixing-key  we 
have,  from  formulas  (49)  and  (50), 


s  =  0.16  -f    l-      =  O.i6  4-  0.7708  =  0.9308" 
and 

st  =  0.16  4-  ^-^  =  0.16  -|-  0.3854  =  0.5454". 

We  may  now  recapitulate  our  dimensions,  and  by  means 
of  Table  IV.  convert  the  decimals  into  convenient 
fractions  :  — 

Diameter,  D  =  36" 

Pitch,  /  =  3^ 

Face  width,  /   =  yf  f" 

Total  tooth  height,  h  =  2ff  " 

Height  below  pitch  circle,  ti  =  iff" 

Height  above  pitch  circle,  h"  —  i£J" 

Breadth  of  tooth  on  pitch  circle,  b   =  i  f" 


154  TOOTHED   GEARING. 

Number  of  teeth,  N  =  29 

Number  of  arms,  n/  •=    4 

Axes  of  arm  cross-sections,       <  7,  ~        ff,, 

( ^      —       2F4 

Thickness  of  rim,  /    =  iff'' 

Nave  length,  I'  =  8JJ" 

Nave  thickness,  k  =  3-^" 

Diameter  of  shaft,  d  =  3fJ" 

Key  width,  s   —  \%"' 

Key  thickness,  s,  =  f  }". 

Fig.  89  shows  the  working  drawings  for  the  above 
spur  wheel.  Fig.  (/?)  is  a  simple  horizontal  projection  of 
the  gear,  showing  the  pitch,  tooth  dimensions,  thick- 
ness of  rim  and  nave,  dimensions  of  arms,  number  of 
teeth,  arms,  etc.  Fig.  (c)  is  a  vertical  projection  taken 
from  Fig.  (b),  as  shown  by  the  dotted  lines,  and  Fig.  (a) 
is  a  sectional,  vertical  projection  taken  from  Fig.  (b)  on 
the  line  AB,  and  showing  the  face  width,  nave  length, 
etc.  The  profiles  were  drawn  by  the  method  of  §  IV., 
Fig.  26. 

Example  2.  —  Required  to  design  and  make  full  work- 
ing drawings  for  a  pair  of  cast-iron  bevel  wheels  to 
transmit  a  force  of  lO-horse  power  from  a  smoothly 
running  turbine  wheel  (moderate  shock),  the  smaller 
bevel  to  be  fixed  upon  the  3"  shaft  of  the  turbine  wheel, 
which  makes  30  revolutions  per  minute,  the  bevel 
wheels  to  be  15"  and  30"  diameters.  The  circumfer- 
ential velocity  of  the  smaller  bevel  (as  also  that  of  the 
larger)  is 

^o  x  TT  x  is      ^o  x  47.124 
v  =  — -  =  —  —  =  2  feet  per  second  nearly. 

12    X    60  12    X    60 


TOOTHED   GEARING. 
ttg.89 


155 


fe) 


*  The  scale  of  all  working  drawings  should  be  £,  |,  i,  -rV,  TjV,  etc.    The 
scale  of  -43U-  is  taken  here  in  order  to  bring  the  drawings  of  convenient  size. 


1  56  TOOTHED   GEARING. 

For  the  smaller  bevel,  from  formula  (14,  b),  we  have, 
therefore,  for  the  pitch, 

/=  i..i7y  ™  =  1.17  x  2.236  =  2.616". 

For  the  face  width, 

/=  2  x  2.616  =  5.232". 

For  the  total  height  of  the  teeth, 

h  =  0.7  x  2.616  =  1.8312". 
For  the  heights  below  and  above  the  pitch  circle, 

h'  '  =  0.4  X  2.616  =  1.0464" 
and 

#'=0.3  x  2.616  =  0.7848". 

Taking,  for  the  breadth  of  the  teeth  at  the  pitch  circle, 
b  =  0.48^,  we  have 

b  =  0.48  x  2.616  =  1.25568". 

The  bevel,  being  so  small,  may  be  made  without  rim 
or  arms,  i.e.,  cast  solid,  as  shown  in  the  drawing  (Fig. 
91,  a).  From  formula  (3)  the  number  of  teeth  is 


2.616 
For  the  thickness  of  the  nave,  from  formula  (35), 

k  =  0.4V2.622  x  7i  4-  \  =  2". 

From  formula  (36),  for  the  length  of  the  nave,  we  have 
/'  =  5.232  -Hi  =  5.732"- 


TOOTHED   GEARING. 


157 


The  diameter  of  the  shaft  is  that  of  the  turbine,  or 
d  =  3".  From  formulas  (49)  and  (50)  the  mean  width 
of  the  key  which  fixes  the  bevel  to  its  shaft  is 


0.16  +  1  =  0.76" 


and  the  thickness, 


=  o.i  6 


=  0.46". 


For  the  larger  bevel  the  pitch  and  tooth  dimensions 
are  the  same  as  for  the  smaller  bevel.  From  formula 
(3)  the  number  of  teeth  is 

TTX  30  _  94.25  _ 
"  2.616   ™  2.616  ~3°' 

From  formula  (34)  the  thickness  of  the  rim  is 

*=  0.12  -f-  0.4  x  2.616  =s  1.1664". 
Formula  (23)  gives  for  the  number  of  arms, 

ns'=  0.56^36^2.616  =  4. 
For  the  number  of  revolutions  per  minute,  we  have, 

Fig.90 


from  formula  (7),  n  =  15.  For 
the  flanged  cross-sections  of  the 
arms,  such  as  that  represented 


in  Fig.  90,  taking  b,,  equal  to  the 
rim-thickness  =  1.1664",  ////  =  i", 
and  B  =  //"',  we  have,  from  for- 
mula (27), 


1.1664  X  fff*  +  H*  X  i  _  126  X  io 
H1  15X4 


158  TOOTHED    GEARING. 

or 

1.1664^3  +  1   =   21. 

Hence 


and 

B  =  Hf  =4.141". 

For  the  thickness  of  the  nave,  from  formula  (35)  we 
have  _ 

k  =  0.4V2.622  X  15  +  i  =  2.36". 

Formula  (36)  gives,  for  the  length  of  the.  nave, 
/'=  5-232+18  =  6.232". 

For  the  diameter  of  the  wrought-iron  shaft  we  have, 
from  formula  (43), 

,/=  3.422^4  =  3". 

Formulas  (49)  and  (50)  give,  for  the  mean  width  and 
thickness  of  the  fixing-key, 

j  =  o.i6  +  §  =0.76" 
and 

/  =  o.i  6  +  T3o  =  0.46". 

Our  dimensions  in  fractions  instead  of  decimals  are  as 
follows  :  — 

For  smaller  bevel. 

Diameter,  D  =  15" 

Pitch,  /  =    2ft" 

Face  width,  /    =    sjf" 

Total  height  of  teeth,  h  =  iff" 
Height  below  pitch  circle,  h'  =  i&" 
Height  above  pitch  circle,  h"  —  f  }" 


TOO  THED    GEA RING. 


159 


Breadth  on  pitch  circle, 
Number  of  teeth, 
Thickness  of  nave, 
Length  of  nave, 
Diameter  of  shaft, 
Key  width, 
Key  thickness, 


For  smaller  bevel. 

b    =    itf" 

N  =  18 
k    =    2" 

i'  =  sir 

d  =    3" 

s   =     ¥' 


For  larger  bevel. 

D  =  30- 


A   =    i 


Diameter, 

Pitch, 

Face  width, 

Total  height  of  teeth, 

Height  below  pitch  circle, 

Height  above  pitch  circle, 

Breadth  of  teeth  at  pitch  circle,  b    =    i  JJ" 

Number  of  teeth,  N  =  36 

Rim  thickness,  t    =    iJJ" 

Number  of  arms,  »/=    4 

(ff=  4  A" 

Arm  dimensions.   See  Fig.  90. 


B  =    4&" 

*»-    t«" 

/<„=    i" 


Thickness  of  nave, 
Length  of  nave, 
Diameter  of  shaft, 
Key  width, 
Key  thickness, 


/'  =  6H" 

</  =  3" 

*  =  r 

/  =  «". 


Fig.  91  gives  the  working  drawings,  drawn  to  a  scale 
of  -f-Q.  Fig.  (<?)  is.  a  sectional  drawing  of  both  bevels 
in  gear,  showing  teeth,  rim,  nave  thickness,  etc.,  and  at 
x  the  true  form  of  the  profiles  and  true  tooth  dimen- 
sions. Fig.  (b)  is  a  partial  projection  of  the  smaller 


i6o 


TOOTHED   GEARING. 


bevel;   and  Fig.  (r),   a  projection  of  the  larger  bevel, 
showing  the  arms,  fixing-key,  etc. 


Example  3.  —  Required  to  design,  and  make  complete 
working  drawings  for,  a  worm  and  wheel  to  transmit  a 


TOOTHED   GEARING.  l6l 

force  of  850  pounds,  little  or  no  shock,  the  wheel  to  be 
12"  in  diameter.  From  formula  (12,  c)  we  have,  for  the 
pitch, 

p  =  0.035^850  =  0.035  x  29-I5  =  1.02", 

for  the  heights  of  the  teeth, 

h  —  0.7  x  1.02  =  0.714" 

h'  =  0.4  x  1.02  =  0.408'' 
and 

h"  =  0.3  x  1.02  =  0.306". 

The  breadth  of  the  teeth  at  the  pitch  circle  is 

b  —  0.48  x  1.02  =  0.4896". 
For  the  number  of  teeth  in  the  wheel,  from  formula  (3), 

JV=^  =  37. 

1.02 

Face  width  of  wheel, 

/=  2  x  1.02  =  2.04". 

From  formula  (23)  the  number  of  arms  is 

n'  =  0.56^37  v/i.02  =  0.56  x  6.08  x  1.005  =  4' 
For  the  thickness  and  width  of   the  arms*  we    have, 
from  formula  (17),  taking  l\  =.  — , 

h?  _  850  X  6  _  1275 

2   ~~  500  x  4  ~~~  500 

*  Ordinarily  so  small  a  gear  would  be  made  without  arms.  For 
the  purpose  of  illustrating,  however,  we  use  four  arms,  as  given  by  the 
formula. 

"of  \ 


1  62  TOOTHED   GEARING. 

or 

*,_VsS-i.7«" 

=  -^  =  0.86". 


2 

Formula  (34)  gives,  for  the  thickness  of  the  rim, 

/=  0.12  4-  0.408  =  0.528". 
From  formula  (35)  the  thickness  of  the  nave  is 

k  =  0.4^1.  022  x  6  +  i  =  0.4  X  1.841  -f  |-  =  1.236". 

The  length  of  the  nave  is,  from  formula  (36), 
/'=  2.04  -f-  J-g-  =  2.44". 

Formula  (39)  gives,  for  the  diameter  of  the  wrought-iron 
shaft  of  the  wheel, 


d  —  o.o86'V85o  x  6  =  0.086  X  17.29  =  1.48". 

From  formulas  (49)  and  (50)  the  width  of  the  fixing- 
key  is 

,  =  0.16  +  ^?  =  0.456" 

and  the  thickness 

/  =  0.16+^  =  0.308". 

From  §  VIII.,  taking  the  radius  of  the  worm  equal  to 
I  \  times  the  pitch,  we  have 

/r=iix  1.02  =  1.53" 
and,  for  the  angle  (X)  of  the  teeth, 

1.02 

tan  A  =  0.159 =  0.159  X  0.6667  =  0.106 


TOOTHED    GEARING.  163 

or  X  =  6°  3  .     From  formula  (39)  the  shaft  diameter  for 
the  worm  is 

d  —  0.086^850  x  1.53  =  0.939". 


Dimensions. 

Diameter  of  wheel,  D  —  12" 

Pitch, 

Total  height  of  teeth, 

Height  below  pitch  circle,  ti  —  -|f 
Height  above  pitch  circle, 

Breadth  of  teeth  on  pitch  circle,  b    =  §£- 

Face  width,  /    =  2-fa 

Number  of  teeth  on  wheel,  N  =  37 

Number  of  arms  on  wheel,  «/  =  4 

Thickness  of  rim,  /    =  -J  J 

Width  of  arms,  h,  =  iff 

Thickness  of  arms,  bt  =  f  f 

Thickness  of  nave,  k    =  iJJ 

Length  of  nave,  /'  =  2T 

Diameter  of  shaft,  d   =  if 

Width  of  key,  s    =  |J 

Thickness  of  key,  /   =  J{ 

Radius  of  worm,  R'  =  i  if 

Angle  of  the  teeth,  A    =  6°  3 

Shaft  diameter  for  worm,  d   =  -JJ 


// 


The  working  drawings,  with  dimensions,  are  given  in 
Fig.  92,  of  which  Fig.  (b)  xis  a  full  projection,  showing 
the  arms,  rim,  nave  thickness,  tooth  dimensions  in  sec- 
tion, angle  (A.) 'of  inclination  of  the  teeth,  etc.  Fig.  (c)  is 
a  sectional  projection  of  Fig.  (b),  showing  the  shape  of 
the  wheel  teeth,  arm  thickness,  nave  length,  etc.  ;  and 
Fig.  (a)  is  a  full  projection  taken  from  Fig.  (b). 


1 64 


TOOTHED   GEARING. 


Example  4.  —  Required  to  design,  and  make  full  work- 
ing drawings  for,  a  pair  of  screw  gears  to  transmit  a 
force  of  2-horse  power,  little  or  no  shock ;  the  larger 


gear  to  be  fixed  upon  a  i  J"  wrought-iron  shaft,  which 
makes  20  revolutions  per  minute,  and  the  smaller  gear 
to  make  40  revolutions  per  minute.  The  angle  included 
between  the  two  gear  shafts  to  be  60°. 


TOOTHED   GEARING.  165 

Suppose  we  take,  for  the  diameter  of  the  smaller 
gear,  6":  hence,  from  formula  (7),  the  diameter  of  the 
larger  gear  is  1 2". 

The  circumferential  velocity  is 

irDn         37.7  x  20 

"  =  17*15  =   12x60  ==  '-°47  feet  Per  second- 

From  §  VIII.  we  have  for  the  angles  of  inclination 
(<£  and  <£')  of  the  teeth,  the  angle  (®)  included  between 
the  axes  of  the  shafts  being  60°, 

<£  +  $  +  0  =  180,     </>  +  <£'  =  180°  -  60°  =  120°. 
If  we  assume  <f>  =  60°,  we  have, 

^=  1 20° -60°  =60°.* 

For  the  larger  wheel  the  dimensions  are  calculated  as 
follows  :  The  pitch,  from  formula  (14,  c),  is 

/  =  o.82\/  — ^—  =  0.82^91  =  0.82  x  1.382  =  1.133". 
?  1.047 

The  face  width  is 

/=  2  x  1.133  =  2.266". 
The  heights  of  the  teeth  are 

h  =  0.7  x  1.133  =  0.793" 

^'=  0.4  x  1.133  =  °-4532" 
and 

A"  =0.3  X  1.133  =  0.3399". 

*  We  can  assume  0  —  90°,  in  which  case  the  gear  upon  which  the 
inclination  of  the  teeth  is  <j>  =  90°  is  a  spur  wheel,  and  then  have 
f  =  120°  —  90°  =  30°  for  the  inclination  of  the  teeth  of  the  other  gear. 


1  66  TOOTHED   GEARING. 

For  the  breadth  of  the  teeth  at  the  pitch  circle  we  may 
take 

b  =  o.48/  =  0.48  x  1.133  =  0.54384"- 

From  formula  (3)  we  have,  for  the  number  of  teeth, 

Ar_   37-7   _ 

~' 


Formula  (34)  gives,  for  the  rim  thickness, 

/=  0.12  +   (0.4   X    I.I33)   =  0.5732". 

From  formula  (35),  the  thickness  of  the  nave  is 


k  =  0.4VI.I332  X  6  +  i  =  0.4V7.70  +  \ 

=  0.4  x  i.97-f|=  1.29". 

Formula  (36)  gives,  for  the  nave  length, 
/'  =  2.266  +  j#=  2.666". 

The  fixing-key  width  and  thickness  are,  from  formulas 
(49)  and  (50), 

s  =  0.16  -f-  -—  =  0.41" 
and 

/=o.i6  +  — =  0.285". 

The  gear  is  small  enough  to  be  made  without  arms. 
The  thickness  of  the  web  between  the  nave  and  rim 
may  be  calculated  from  formula  (24),  by  assuming  the 
gear  to  have  10  arms,  the  width  of  each  being  one- 
tenth  the  outer  circumference  of  the  nave.  Thus  the 


TOOTHED    GEARING.  1 67 

shaft  diameter  is  1.2$",  and  the  nave  thickness  1.29": 
hence  the  diameter  across  the  nave  is 

1.25  -f  (2  x  1.29)  =3.83" 

and  the  circumference  12.052".  The  width  of  the 
assumed  arms  is  therefore  -  '— — ,  or//,  =  1.203".  For- 
mula (24)  becomes 

b,  x  i.2O32 
or 

*,  = -^  =  0.87". 
1.447 

For  the  smaller  gear  the  pitch  and  tooth  dimensions 
are  the  same  as  for  the  larger  gear,  as  is  also  the  rim 
thickness.  The  thickness  of  the  nave  is,  from  formula 
(35), 


k  =  o.4Vi.i332  X  3  +  J  =  o.4V3.85i  +  J 

=  0.4  X  1.567  +  I  =  I.I268". 

From  formula  (36)  we  have,  for  the  length  of  the  nave, 

/'=  2.2664-  &  =  2.466". 
From  formula  (3),  for  the  number  of  teeth,  we  have 

18.85 

N= 5.=  17. 

i-i33 

The  diameter  of  the  wrought-iron  shaft  is,  from  for- 
mula (43), 

<t=  3.42 2^£  =  3.422^.050  =  3.422  x  0.368  =  1.2593". 


and 


1  68  TOOTHED   GEARING. 

Formulas  (49)  and  (50)  give,  for  the  mean  width  and 
thickness  of  the  fixing-key, 


=  0.41186 


/=  o.i  6  +  =  0.28593". 

For  the  thickness  of  the  web  between  the  rim  and 
nave,  the  diameter  across  the  nave  is 

i.  2593  +  (2  x  1.1268)  =  3.51" 

and  the  circumference  11.03":  hence  //,  —  i.io".     And 
formula  (24)  gives 

126  x  2 


40  x  10 
or 


bl  X  i.io2  = 
b 

• 


Dimensions  for  larger  gear. 

Diameter,  d  =12" 

Pitch,  /  =    i&" 

Face  width,  /    =    2}%' 

Total  height  of  teeth,  h  =  -|J" 
Height  below  pitch  circle,  h'  =  ff  " 
Height  above  pitch  circle,  h"  —  %%f 
Breadth  at  pitch  circle,  b  =  ff  " 
Rim  thickness,  /  =  ff" 

Number  of  teeth,  N  =  33 

Thickness  of  web,  bl  =      |" 

Thickness  of  nave,  k    =    i^/' 

Length  of  nave,  /'  =    2|f  " 


TOOTHED   GEARING.  169 

Dimensions  for  larger  gear. 

Diameter  of  shaft,  d  =    ij" 

Width  of  fixing-key,  s  —      -J-f  " 

Thickness  of  fixing-key,  /  =      •£%" 

Angle  of  the  teeth,  <£  =  60°. 

Dimensions  for  smaller  gear. 

Diameter,  D  =  6" 

Pitch,  /  =  i^" 

Face  width,  /    =  2^-J" 

Total  height  of  teeth,  h    =  f£" 

Height  below  pitch  circle,  h'  —  ff" 

Height  above  pitch  circle,  //'  =  J£" 

Breadth  of  teeth,  b    =  f  f " 

Rim  thickness,  /    =  JJ" 

Number  of  teeth,  N  =  1 7 

Thickness  of  web,  bl  =  f£" 

Thickness  of  nave,  k    —  i£" 

Length  of  nave,  /'  =  2  Jf " 

Diameter  of  shaft,  //   =  i  J" 

Width  of  fixing-key,  s    =  £f " 

Thickness  of  fixing-key,  /   =  ^" 

Angle  of  the  teeth,  <£'  =  60°. 

The  working  drawings  with  marked  dimensions  are 
given  in  Fig.  92  A.  Fig.  (a)  is  a  full  projection  of  the 
larger  gear,  showing  the  pitch,  tooth  dimensions,  rim, 
etc.  The  right  half  of  Fig.  (b)  is  a  full  projection  of 
the  larger  gear,  taken  from  Fig.  (a) ;  and  the  left  half 
is  a  sectional  projection  taken  from  Fig.  (a),  showing 
the  web  thickness,  etc.  Similarly,  for  the  smaller  gear, 
Fig.  (c)  is  a  full  projection,  and  Fig.  (d)  a  full  and  sec- 
tional projection  taken  from  Fig.  (c). 

Example  5.  —  Required  to  design,  and  make  complete 
working  drawings  for,  a  cast-iron  internal  spur  gear  and 


I/O 


TOOTHED   GEARING. 


pinion  which  will  safely  transmit  a  force  of  6,197  pounds 

Fig.  92  A 


SCALE  = 


moderate  shock,  the  pinion  to  be  fixed  upon  a 
wrought-iron  shaft,  the  face  width  to  be  2j  times  the 


TOOTHED   GEARING.  I/I 

pitch,  and  the  revolution  ratio  3  to  i.     From  formula 
(n,  b)  we  have,  for  the  pitch, 


/  =  o.o7y  6197  x  ~  =  0.07^2478.80  =  0.07x49.8  =  3.486". 

For  the  face  width, 

/=2^  x  3.486  =  8.715". 
The  heights  are, 

h  —  0.7  x  3.486  =  2.44" 

//=o.4  x  3486  =  1.394" 
and 

#'  =  0.3  x  3.486  =  1.046". 

Taking  the  breadth  of  the  teeth  equal  to  0.45^,  we  have 
£  =  0.45  x  3486=  1.569". 

If  we  take  for  the  diameter  of  the  pinion  25^",  we  shall 
have  for  the  number  of  teeth,  from  formula  (3), 

80.10 


From  formula  (7)  the  diameter  of  the  internal  gear  is 

25i  X  3  =  76i"> 
and  from  formula  (3)  the  number  of  teeth  is 

=  69. 


1 72  TOOTHED   GEARING. 

Formula  (34)  gives,  for  the  rim  thickness  of  the  pinion, 
/=  0.12  +  (0.4  x  3.486)  =  1.51". 

Since  in  an  internal  gear  the  rim  is  not  supported  by 
the  arms,  as  in  an  external  gear  (see  Fig.  93,  #),  we  may 
take  the  rim  thickness  for  the  internal  gear  equal  to 
2t  =  3".  From  formula  (35),  the  thickness  of  the  nave 
for  the  pinion  is 


k  =  o.43v/3.4862  x  12.75  +  \  =  0.4^/12.15  X  12.75  4-  } 

=  0.4  x  5.371+^=2.6484", 
and,  for  the  internal  gear, 


k  =  o.4'V3.4862  x  38.25  +  \  =  0.4^12.15  x  38.25  + 1 

=  0.4  x  7.746+1=  3.598". 

Formula  (36)  gives  for  the  nave  lengths  of  the  pinion 
and  internal  gear  respectively, 

/'  =  8.715  +  2-~  =  8.715  +  0.85  =  9.565" 
and 

/'=  8.715  +  ^  =  8.715  +  2.55  =  11.265". 

The  pinion  may  be  without  arms,  and  the  thickness  of 
the  web  calculated  from  formula  (29)  by  assuming  the 
pinion  to  have  10  arms,  each  having  a  width  of  one- 
tenth  the  outer  circumference  of  the  nave.  Thus  the 
diameter  of  the  shaft  is  3.6875",  and  the  nave  thickness 
2.6484":  hence  the  diameter  across  the  nave  is 

3.6875  +  (2  x  2.6484)  =  9", 


TOOTHED   GEARING. 


and  the  circumference  28.27".  We  therefore  have 
//,  =  2.827";  ancl  formula  (29)  gives 

0.80  x  3.486*  x  12.715 
^X2.8272=-          ^_  ^=12.393, 

or 

12,393  „ 

7-99 

For  the  number  of  arms  for  the  internal  gear,  formula 
(23)  gives 

n!  =  0.56^  >/3.486  =  0.56  x  8.307  x  1.366  =  6.35, 

say  ;//  —  7.  If  we  wish  to  have  elliptical  arm  cross- 
sections,  we  have  from  formula  (31),  taking  the  minor 
axis  equal  to  one-half  the  major, 

03  3.486*  x  38.25 

*,«»  =  -  =  1.356  6-     —^      *  =  90.03. 

Hence  _ 

a  =  V9°-°3  x  2  =  5.647" 
and 

*I  =  S^47  =  2i8    „. 


From  formula  (39),  the  diameter  of  the  wrought-iron 
shaft  for  the  internal  gear  is 


</=  o.o86V6i97  X  38.25  =  0.086  X  61.888  =  5.322". 

Formulas  (49)  and  (50)  give,  for  the  mean  width  and 
thickness  of  the  fixing-key  for  the  pinion, 


s  =  0.16  +  =  0.16  +  0.7375  =  0-8975" 


«  o.i6  +  °'   Q'J  =  0.16  +  0.36875  =  0.52875' 


1/4  TOOTHED   GEARING. 

and  the  same  for  the  internal  gear, 


*  =  o.i6  +  ^^=1.2244" 

and 

s'  =  0.16  +  —  -  =  0.6922". 

Dimensions  for  pinion. 

Diameter,  D  =  25!" 

Pitch,  /  =  3fA" 

Face  width,  /    =  8f|" 

Total  height  of  teeth,  h    =  2Ty 

Height  below  pitch  circle,  ti  =  if f" 

Height  above  pitch  circle,  h"  —  i^" 

Number  of  teeth,  N  =  23 

Breadth  of  teeth,  b    =  iTy 

Thickness  of  rim,  /    =  iff" 

Thickness  of  nave,  k    =  2%%" 

Length  of  nave,  /'  =  9Ty 

Thickness  of  web,  b,  =  iff" 

Diameter  of  shaft,  d   =  3^" 

Mean  width  of  fixing-key,  s    =  f  §" 

Thickness  of  fixing-key,  /  =  | " 

Dimensions  for  internal  gear. 

Diameter,  D  =  76^" 

Pitch,  /  =    3U" 

Number  of  teeth,  N  =  69 

Total  height  of  teeth,  h  —  2TV' 
Height  below  pitch  circle,  h'  =  iff'' 
Height  above  pitch  circle,  h"  =  i$$" 
Face  width,  /  =  8ff" 

Breadth  of  teeth,  b    =    i^" 

Rim  thickness,  2/  =    3" 

Nave  thickness,  k    = 


TOOTHED   GEARING.  175 

Dimensions  for  internal  gear 

Nave  length,                                 /'  =  n£J" 

Number  of  arms, 

Major  axis  of  arm  cross-sections,  a 

Minor  axis  of  arm  cross-sections,  //  =  2ff " 

Diameter  of  shaft,                          d  —  5§  J" 

Width  of  fixing-key,                      s  =  ifa" 

Thickness  of  fixing-key,               /  =  JJ" 

Fig.  93  shows  the  working  drawings  for  the  pair  of 
gears,  to  the  scale  of  -^  ;  Fig.  (a)  being  a  full  projection 
of  both  gears  in  position  for  action,  showing  the  pitch, 
tooth  dimensions,  number  of  arms,  etc.,  and  Fig.  (b) 
being  a  sectional  projection  of  both  gears,  taken  from 
Fig.  (a),  on  the  line  xy,  showing  the  shape  of  the  arms 
of  the  larger  gear  necessary  to  the  proper  action  of  the 
pair,  etc. 

Example  6.  —  Required  to  design,  and  make  full 
working  drawings  for,  a  cast-iron  rack  and  pinion  to 
transmit  a  force  of  1,000  pounds,  moderate  shock ;  the 
pinion  to  make  20  revolutions  per  minute,  and  the  rack 
(which  is  to  be  9  feet  long)  to  move  at  the  rate  of  63  \ 
feet  (762")  per  minute.  From  formula  (12,  b),  for  the 
pitch  we  have 

p  =  0.05^1000  =s  0.05  x  31.62  ==  1.581". 
The  face  width  is,  consequently, 

/=  2  x  1.581  =  3.162". 
For  the  heights  of  the  teeth, 

h  —  0.7  x  1.581  =  1.1067" 

W '=  0.4  x  1.581  =  0.6324" 
and 

#'  =  0.3  x  1.581  =  0.4743". 


TOOTHED   GEARING. 


Taking  the  breadth  of  the  teeth  equal  to  0.48  times  the 
pitch  gives 

b  =  0.48  x  1.581  =  0.7589". 


The  circumferential  velocity  of  the  pinion  (which  is  equal 
to  the  velocity  of  the  rack)  is  762"  per  minute  :  hence 
the  circumference  of  the  pinion  must  be  J2^=38.i". 


TOOTHED   GEARING.  If? 

From  formula  (3),  the  number  of  teeth  is 


and  from  formula  (2),  the  diameter  is 


3-I4I59 

Formula  (34)  gives,  for  the  thickness  of  the  rim, 

/=  0.12  -f  (0.4  x  1.581)  =  0.7524", 
and,  from  formula  (35),  the  nave  thickness  is 


k  =  o.4Vi.58i2  x  6.0625  +  -J  =  o.4Vi5.i56  +  £ 

=  0.4  x  2.475  +  i  =  M9 

The  nave  length  is,  from  formula  (36), 

e  =-162  =  .66". 


From  formula  (39)  we  have,  for  the  diameter  of  the 
wrought-iron  pinion  shaft, 

d=  0.086  AOOO  x  6.0625  =  o.o86'V6o62.5 

=  0.086  X  18.234  =  1.5  f. 

Formulas  (49)  and  (50)  give,  for  the  width  and  thickness 
of  the  pinion  fixing-key, 

*  =  0.16  +  ^  =  0.474" 
and 

/«  0.16  + 


178  TOOTHED   GEARING. 

The  pinion  is  small  enough  to  be  made  without  arms. 
For  the  thickness  of  the  web  we  have  the  following. 
The  diameter  across  the  nave  is 

1.5 7  +  (2  x  1.49)  =4-55"> 
and  the  circumference  is 

4.55  x  3.14159=-=  14.294". 
Hence  //,  =  1.43",  and  formula  (17)  becomes 

1000  x  6.0621% 

bth?  =  2.045^,  =  - 

500  x  10 

1.2  I  2 S  „ 

bl  — =  0.599". 

2.045 

The  dimensions,  converted  into  fractions,  are  as  fol- 
lows :  — 

Diameter  of  pinion,  D  =  i2|" 

Length  of  rack,  9' 

Pitch,  /  =    iff" 

Number  of  teeth,  N  =  24 

Face  width,  /    =    3&" 

Total  height  of  teeth,          h   =    !•&" 
Height  below  pitch  circle,  h'  —      f" 
Height  above  pitch  circle,  h"  —      £f " 
Breadth  of  teeth,  b   =      ft" 

Thickness  of  rim,  /    =      f " 

Thickness  of  nave,  k  —    i£" 

Length  of  nave,  /'  = 

Diameter  of  shaft,  d  =; 

Width  of  key,  s    — 

Thickness  of  key,  /  = 

Thickness  of  web,  £,  = 


TOOTHED   GEARING. 


179 


The  working  drawings  are  shown  in  Fig.  94,  drawn 
to  a  scale  of  T36-,  and  dimensions  marked.     Fig.  (a)  is  a 


full  projection  of  the  rack  and  pinion  in  gear,  the  rack 
being  broken  in  order  to  save  space ;  Fig.  (c),  a  full 
projection  taken  from  Fig.  (a) ;  and  Fig.  (#),  a  sectional 


180  TOOTHED   GEARING. 

projection  of  the  rack  and  pinion,  taken  from  Fig.  (a), 
on  the  line  xy.  The  cycloidal  profiles  of  the  teeth  were 
drawn  by  the  method  given  under  Fig.  26  for  the  pinion, 
and  under  Fig.  35  for  the  rack. 

Example  7.  —  Required  to  design,  and  make  working 
drawings  for,  a  cast-iron  lantern  gear  and  pinion  to 
transmit  a  force  of  1,600  pounds,  moderate  shock,  the 
revolution  ratio  of  the  lantern  to  the  pinion  being  £. 

From  formula  (12,  b),  for  the  pitch, 

p  =  0.05^1600  =  0.05  x  40  =  2". 
The  total  height  of  the  teeth  is 

h  =  0.7  x  2  =  1.4", 
and  the  breadth  may  be 

b  =  0.46  x  2  =  0.92". 
The  face  width  is 

1=2    X    2=4". 

If  we  take,  for  the  diameter  of  the  lantern,  19^",  we 
have  for  the  number  of  teeth,  from  formula  (3), 

60.08 
;V=-— -  =  3o 

and,  from  formula  (7),  the  diameter  of  the  pinion  is 

a*-6r. 

The  number  of  teeth  for  the  pinion  is 

20.02 
=  10. 

2 

Formula  (34)  gives,  for  the  rim  thickness, 
/=  0.12  -f-  (0.4  x  2)  =  0.92". 


TOOTHED   GEARING.  l8l 

Formula  (35)  gives  for  the  nave  thickness,  for  the  lan- 
tern, 


k  =  0.41/2-  x  9.5625  +  |  =  0.4-^38.25  +  i 

=  0.4  x  3.369  +  £=  1.8476", 
and  for  the  pinion, 


k  =  o.4'V22  X  3.1875  -f-  1=  o.4'Vi2.75  -h  | 

=  0.4  x  2.336  +  1  =  1.4344"- 

For  the  nave  length  of  the  pinion  we  have,  from  for- 
mula (36), 


and  for  the  lantern, 

r-  4  +  ^  =  4.6375". 

The  diameter  of  the  pinion  shaft,  from  formula  (39),  is 

</=  o.o86Vi6oo  x  3.1875  =  0.086^5100 

=  0.086  X  17.213  =  1.48", 

and  the  diameter  of  the  lantern  shaft, 

d—  o.o86Vi6oo  x  9.5625  =  o.o86Vi53QO 

=  0.086  X  24.826  =  2.135". 

For  the  pinion,  the  width  and  thickness  of  the  fixing- 
key,  from  formulas  (49)  and  (50),  are 

s  =  0.164-  —  =  0.456" 
and 


0.164-          =  0.308". 


1 82  TOOTHED   GEARING. 

For  the  lantern, 

s  —  0.16  +  ^12$  =  0.587" 

3 

and 

/=  o.i  6  +  2-^  =  0.3735". 


The  pinion  is  small  enough  to  be  made  without  arms. 
Formula  (23)  gives,  for  the  number  of  arms  in  the  lan- 
tern, 

n;  =  0.56^30  V2  =  0.56  x  5.48  x  1.19  =  4. 

For  arms  having  circular  cross-sections,  formula  (19) 
gives  a  diameter  of 

4  3/1600x0. 562^  3/—r — 

d  =0.15^  -          v:>     *  =  0.15^/3825  =  0.15  x  15.64=  2.346". 
4 

As  explained  in  §  VI.,  under  Fig.  42,  the  radius  for  the 
lantern  rungs  is  j$  X  2  =  0.475". 

Dimensions  for  the  lantern. 

Diameter,  D  =  ipj" 

Pitch,  /  =  2" 

Face  width,  /    =  4"* 

Radius  of  rungs,  =      Jf" 

Number  of  rungs,  N  =  30 

Thickness  of  rim,  /    =      fj" 

Number  of  arms,  «/  =  4 

Diameter  of  arm  cross-section,  dr  =  2  JJ" 

Thickness  of  nave,  k    =  iff" 

Length  of  nave,  /'  =  4f£" 

Diameter  of  shaft,  d  =  2^" 

Width  of  fixing-key,  j    =      J|" 

Thickness  of  fixing-key,  /  =      f " 

*  See  Fig.  95  (c). 


TOOTHED   GEARING.  183 

Dimensions  for  the  pinion. 

Diameter,  D  =  6f " 

Pitch,  /  =  2" 

Face  width,  /    =  4" 

Total  height  of  teeth,  h    —  i£f" 

Breadth  of  teeth,  b    =  f  J" 

Number  of  teeth,  JV  —  10 

Thickness  of  nave,  k    —  i-fr" 

Length  of  nave,  /'  = 

Diameter  of  shaft,  d  = 

Width  of  fixing-key,  s    = 

Thickness  of  fixing-key,  /   =  T5/'. 

Fig.  95  gives  the  working  drawings  of  the  lantern  and 
pinion,  drawn  to  a  scale  of  -£%-  One  of  the  lantern  rungs 
is  shown  in  section  in  Fig.  (c)  in  order  to  show  that  the 
rungs  are  to  be  cast  on  the  lantern,  instead  of  being 
made  separately,  and  driven  into  holes  along  the  lantern 
rim,  as  is  ordinarily  the  case.  The  arrangement  of  the 
rim,  etc.,  is  sufficiently  explained  by  the  figure.  The 
teeth  of  the  pinion  are  drawn  according  to  the  explana- 
tion given  in  §  VI.,  under  Fig.  42. 

Example  8.  —  Given  the  data  and  dimensions  of  the 
pinion  of  Example  7,  it  is  required  to  design  an  internal 
lantern,  the  revolution  ratio  of  which  to  the  pinion  shall 
be  \ ;  the  rungs  of  the  lantern  to  be  of  wrought-iron, 
and  to  be  driven  into  holes  along  the  rim. 

The  radius  for  the  rungs  is  the  same  as  in  Example  7, 
as  is  also  the  calculated  rim  thickness.  But  for  an  in- 
ternal gear  we  take  the  rim  thickness  from  ij  times  to 
twice  that  of  an  external  gear  (see  Fig.  96,  b). 

From  formula  (7),  the  diameter  of  the  lantern  is 

D  =  6§  X  4  =  25 J' ', 


1 84 


TOOTHED   GEARING. 


and,  from  formula  (3),  the  number  of  rungs  is 


..     80.1 
N= =  40. 


Formula  (23)  gives,  for  the  number  of  arms, 

n'  —  0.56^40 \/2  =  0.56  x  6.32  x  1.187  =  5« 


From  formula  (19)  the  diameter  for  the  circular  cross- 
section  of  the  arms  is 


.3/1600X12.75  s/ — r~  „ 

=  o.i5y-     — —     -  =  0.15^4080  —  0.15x15.98  =  2.40. 


TOOTHED    GEARING.  185 

For  the  nave  thickness,  formula  (35)  gives, 

£  =  0.4^2*  x  12.75  +i  =  0-4^5 r  +4  =  0.4x3.7  -f  J  =  1.98", 
and  the  nave  length  is,  from  formula  (36), 
/'=  4  +  2£S  =  4.8S". 

Formula  (39)  gives,  for  the  diameter  of  the  wrought-iron 
lantern  shaft, 


d  —  o.o86'Vi6oo  x  12.75  =  o.o86Y20400  =  0.086  X  27.32 

=  2.349"- 

The  width  and  thickness  of  the  fixing-key  are,  from  for- 
mulas (49)  and  (50), 


and 


=  0.16  +  =  0.395 


" 


Dimensions  for  lantern. 

" 


Diameter,  D  —  25% 

Pitch,  /  =  2" 

Face  width,  /    =  4" 

Radius  of  rungs,  -Jf  " 

Number  of  rungs,  N  =  40 

Thickness  of  rim,  /    =  ff-" 

Number  of  arms,  #/  =  5 

Diameter  of  arms,  d'  =  2\%' 

Thickness  of  nave,  k    =  iff" 

Length  of  nave,  /'   =  4f|" 

Diameter  of  shaft,  d    =  2§f" 

Width  of  fixing-key,  j    =  JJ" 

Thickness  of  fixing-key,  /    =  |f" 


1 86  TOOTHED   GEARING. 

Dimensions  for  pinion. 

Diameter,  D  =  6f" 

Pitch,  /  =  2" 

Face  width,  /    ==  4" 

Total  height  of  teeth,  h    =  i-|f" 

Breadth  of  teeth,  £    =  £f" 

Number  of  teeth,  ^V  =  10 

Thickness  of  nave,  k    =  i-&" 

Length  of  nave,  /'   =  4-^" 

Diameter  of  shaft,  //    =  ifj" 

Width  of  fixing-key,  j     =  f£" 

Thickness  of  fixing-key,  /   =  ^". 

The  working  drawings  for  Example  8  (shown  in  Fig. 
96,  drawn  to  a  scale  of  ^)  need  but  little  explanation. 
The  dimensions  are  marked  on  the  drawings  ;  and  the 
arrangement  of  the  lantern  arms,  proportions  of  the 
rim,  etc.,  will  be  sufficiently  explained  by  a  glance  at 
Fig.  (b).  The  teeth  of  the  pinion  were  drawn  by  the 
method  explained  in  §  VI.,  under  Fig.  44. 

Example  9.  —  Required  to  design  a  train  of  cast-iron 
gears  to  lift  a  weight  of  8,000  pounds  (say,  moderate 
shock)  by  means  of  a  drum  and  cord  as  outlined  in 
Fig.  97- "  The  circumferential  force  of  the  driving-gear  r 
is  1,000  pounds,  and  the  diameter  of  the  driver  12".  Let 
us  assume  that  ten  per  cent  of  the  driving-force  is  lost 
in  overcoming  the  friction  of  the  gear  teeth,  shaft  bear- 
ings, etc.  We  have,  therefore,  an  effectual  force  of 
1000—1000X0.10  =  900  pounds,  with  which  to  lift 
the  weight  of  8,000  pounds.  We  must  gear  our  power 
from  900  pounds  to  8,000  pounds,  or,  in  other  words,  we 
must  gear  our  power  up  -^o0-  =  9  times.  Since  the 
powers  of  the  gears  are  inversely  proportional  to  their 


TOOTHED    CKAKIXG. 


IS7 


radii  (formula  8),  we  must  gear  down  our  radii  9  times. 
We  can  gear  from  R  to  r'  2\  times,  and  from  Rf  to 


the  drum  r"  4  times  (2\  x  4  =  9).      If,  therefore,  we 
take  R  =  13^",  we  have 


1 88 


TOOTHED   GEARING. 


and,  if  we  take  R'  —  28",  we  have,  for  the  radius  of  the 
drum, 


Fig. 97 


The  power  (or  circumferential  force)  of  the  gear  R  is,  of 
course,  that  of  the  driver  r,  1,000  pounds;*  and  from 

formula  (8)  the  power  of 
the  gear  rf  (and  conse- 
quently that  of  the  gear 
R')  is  1000  X  2\  =  2250 
pounds.  The  total  power 
of  the  drum  is  2250  X  4 
=  9000  pounds.  Our  ex- 
ample is  now  reduced  to 
two  very  simple  ones  ; 
viz.,  first  to  design  a  pair 
of  gears  (r  and  R)  to 
transmit  a  force  of  1,000  pounds  (moderate  shock),  the 
diameters  to  be  2r  =  1  2",  and  2R  =•  27"  ;  and,  second, 
to  design  a  pair  of  gears  (r'  and  R')  to  transmit  a  force 
of  2,250  pounds  (moderate  shock),  the  diameters  being 
2r'=i2",  and  2R'  —  $6".  Let  us  take  them  in  the 
order  given.  From  formula  (12,  b),  the  pitch  for  the 
gears  r  and  R  is 

/  =  0.05^1000  =  0.05  x  31.62  =  1.581" 

for  the  face  width, 

/=  2  x  1.581  =  3.162". 


*  We  do  not  take  the  lost  power  into  account  in  calculating  the 
strength  of  the  gears. 


TOOTHED   GEARING.  189 

The  heights  are, 

h  =  0.7  x  1.581  =  1.1067" 

h'  —  0.4  x  1.581  =  0.6324" 
and 

h"  =  0.3  x  1.581  =  0.4743". 

Taking  the  breadth  of  the  teeth  equal  to  0.45^  gives 
b  =  0.45  x  1.581  =  0.7115". 

From  formula  (3),  the  number  of  teeth  for  r  is 

^=37^9 
1.581 
and  for  R, 


Formula  (34)  gives,  for  the  rim  thickness, 

/=  0.12  4-  (0.4  x  1.581)  =  0.7524". 

The  gear  r  is  without  arms.     For  the  gear  R  the  num- 
ber of  arms,  from  formula  (23),  is 

«/  =  0.56^54  Vi.sSi  =  0.56  X  7.348  x  1.  121  =  5. 

For  elliptical  cross-sections,  taking  b'  =  -,  formula  (20) 

gives 

a*  1000  X  13.5 

b'a*  =  ~  =  0.00339  --  —  —  =  9-153 

or 

a  =  Vi8.so6  =  2.636" 


19°  TOOTHED  GEARING. 

Formula  (35)  gives,  for  the  nave  thickness  for  r, 


k  =  o.4Vi.58ia  x  6  +  J  =  o.4Vi5  +  | 

=  0.4  x  2.466  4-  \  —  1.486" 
and  for  R, 


k  =  o.4Vi.58i2  X  13.5  4-  i  =  0.4Y/33-744  +  I 

=  0.4  X  3.231  +  £  =  1.794", 

From  formula  (36),  the  nave  length  for  r  is 

/'=  3.162  4-t»  =  3.562", 
and  for  J?, 

/' =3.1624-1$  =  4-062". 

The  diameter  of  the  shaft  for  r  is,  from  formula  (39), 


d—  0.086^1000  x  6  =  0.086  x  18.17  =  1.5626" 
and  for  R, 


d—  0.086^1000  x  13.5  =  0.086  x  23.81  =  2.0477". 

Formulas  (49)  and  (50)  give,  for  the  width  and  thick- 
ness  of  the  fixing-key  for  r, 

1.^626 
s  =  0.16  H =  0.4725" 

and 


/=  0.16  +  ^^  =  0.3163", 

and  for  R, 

,=0.l6  +  ^  = 

and 


/=  0.164-^^  =  0.3648". 
For  the  thickness  of  the  web  between  the  nave  and 


TOOTHED   GEARING.  191 

rim  of  the  gear  r,  the  calculations  are  as  follows.  The 
diameter  across  the  nave  is  equal  to 

d+  2k  =  1.5626  4-  (2  X  1.486)  =  4.535" 

and  the  circumference  is  14.2$".  Supposing  the  gear  to 
have  10  arms,  each  having  a  width  of  one-tenth  the 
nave  circumference,  we  have 


Formula  (17)  therefore  gives,  for  the  web  thickness, 

1000  x  6 
bji*  —  2.03/>,  =  — 

500  x  10 

or 

1000  x  6  „ 

bl  = —  =  0.591". 

500  X  10  X  2.03 

For  the  second  pair  of  gears,  rf  and  Rf,  formula  (12,  b) 
gives  a  pitch  of 

/  =  o.o5V/l2~50  =  0.05  x  47434  =  2.3717"- 

The  face  width  is 

/=  2  x  2.3717  =  4.7434". 

For  the  heights  of  the  teeth  we  have 

h  =  0.7  x  2.3717  =  1.6602" 

/&'=  0.4  x  2.3717  =  0.9487" 
and 

A"  =0.3  x  2.3717  =  0.7115". 

The  breadth  of  the  teeth  at  the  pitch  circle  is 
^  =  0.45  x  2.3717=  1.0673". 


192  TOOTHED   GEAK1XG. 

From  formula  (3),  the  number  of  teeth  for  r1  is 


and  for  R't 

N  =  115^3  = 
2-3717 

The  small  gear  rf  is  without  arms.    From  formula  (23), 
the  number  of  arms  for  Rf  is 

«/=  0.56^74  V2.37I7  =  0.56  x  8.60  x  1.241  =  6. 

For  elliptical  cross-sections,  taking  tf  =  -,  formula  (20) 
gives 


7,      ,  «3  2250     X      28 

*  *    =    ~     =    0.00339    -     5-_-     -     =     35.60, 


or 


a  =  ^71.20  =  4.1447" 
and 

j,_  4.H47  =        2  ,/ 

2 

The  thickness  of  the  rim  is,  from  formula  (34), 
/=  0.12  -h  (0.4  x  2.3717)  =  1.0687". 

Formula  (35)  gives,  for  the  nave  thickness  for  /, 


k  =  0.4/2.372  x  6  -f  \  =  0.4^33.701  +  £ 

=  0.4  x  3.23  +|=  1.792" 
and  for  the  gear  Rf, 


k  =  o.4V2.372  x  28  +  £  =  0.4^157.273  +  J 

=  0.4  x  5.398  +  1=  2.6592". 

From  formula  (36),  the  nave  length  for  r'  is 
/'=  4.7434  -f  if  =  5.1434", 


TOOTHED   GEARING.  193 

and  for  R\ 

/'=  4-7434  4-18  =  6.61". 

For  the  shaft  diameter  for  /-',  formula  (39)  gives 

d =  0.086^2250  x  6  =0.086^13500  =  0.086  x  23.81  =  2.048" 

and  for  R't 

//= 0.086^2250  x  28 =0.086  v  63000  =  0.086  x  39.79  =  3.42 1  9". 

For  the  width  and  thickness  of  the  fixing-key  for  r'« 
formulas  (49)  and  (50)  give 

,r  =  o.i6  +  -      -  —  0.5696'' 
and 


/=  o.i  6  +  ^^  =  0.3648", 
and  for  R't 


and 


10 


For  the  web  thickness  for  ;-',  as  before,  the  nave 

diameter  is 

d  +  2k  =  2.048  +  (2  x  1.792)  =  5.63", 

and  the  circumference  is  17. 69":  hence 


10 
From  formula  (17), 

2250  x  6 
i  i   —  3-J3  i  —  ^00  x  I0 

or 


500  x  10  x  3.13 


194  TOOTHED   GEARING. 

Dimensions  for  gear  r. 

Diameter,  D  =  12" 

Pitch,  /  =    ifj" 

Face  width,  /    =    3^" 

Total  height  of  teeth,  h    =    i-fa" 

Height  below  pitch  circle,  h'  —      J|" 

Height  above  pitch  circle,  h"  =      if" 
Breadth  of  teeth  on  pitch  circle,  b    =      if" 

Number  of  teeth,  N  =  24 

Rim  thickness,  /    =      £-" 

Nave  thickness,  k    =    ifj" 

Nave  length,  /'  =    3yV 

Shaft  diameter,  d   =    r^-" 

Width  of  fixing-key,  s    =      ^f  " 

Thickness  of  fixing-key,  /   =      -£$" 

Thickness  of  web,  b,  =      \\ " 

Dimensions  for  gear  R. 

Diameter,  D  =  27" 

Pitch,  p  =    ifj" 

Face  width,  /    =    33%" 

Total  height  of  teeth,  h    —    i^" 

Height  below  pitch  circle,  h'  —      |-J" 

Height  above  pitch  circle,  h"  =      tt" 
Breadth  of  teeth  on  pitch  circle,  b    =      f|" 

Number  of  teeth,  N  =  54 

Rim  thickness,  /    =      f" 

Number  of  arms,  n/  =    5 

Major  axis  of  cross-sections,  a    =    2^" 

Minor  axis  of  cross-sections,  b'  =    i  A" 

Nave  thickness,  £    =    ifj-" 

Nave  length,  /'  =    4Ty 

Shaft  diameter,  d   =    2-^" 

Width  of  fixing-key,  s    =      |f" 

Thickness  of  fixing-key,  /   =      f  J" 


TOOTHED   GEARING.  195 

Dimensions  for  gear  r'. 

Diameter,  D  =  12" 

Pitch,  /  =    2f" 

Face  width,  /    =    4j" 

Total  height  of  teeth,  h    =    if|" 

Height  below  pitch  circle,  h'  —      f  J" 

Height  above  pitch  circle,  h"  =      jj" 

Breadth  of  teeth  on  pitch  circle,  ^    =    i^" 

Number  of  teeth,  N  =  16 

Rim  thickness,  t    =    i^V' 

Nave  thickness,  /£    =    if  J" 

Nave  length,  /'   =    5^:" 

Shaft  diameter,  d   =    2^" 

Width  of  fixing-key,  j    =      fj" 

Thickness  of  fixing-key,  /   =      f f" 

Thickness  of  web,  b^  —      f  }'r 

Dimensions  for  gear  /?'. 

Diameter,  Z>  =  56" 

Pitch,  /  =    2|" 

Face  width,  /    =    4j" 

Total  height  of  teeth,  h    —    ifj" 

Height  below  pitch  circle,  #  =      f  J" 

Height  above  pitch  circle,  //'  =      ^}" 
Breadth  of  teeth  on  pitch  circle,  b    —    i^" 

Number  of  teeth,  N  =  74 

Rim  thickness,  /    =    i^r" 

Number  of  arms,  n{  —    6 

Major  axis  for  cross-sections,  a    —    4^" 

Minor  axis  for  cross-sections,  b'  =    2-faf' 

Nave  thickness,  k    —    2%%" 

Nave  length,  /'   =    6}|" 

Shaft  diameter,  d   = 

Width  of  fixing-key,  s    — 

Thickness  of  fixing-key,  /   = 


I96 


TOOTHED   GEARING. 


The   working   drawings  for  the    train    are   given    in 
Fig.  98,  drawn  to  a  scale  of  g3^.     Fig.  (a)  is  a  full  projec- 


tion of  the  whole  train,  showing  the  pairs  in  gear ;  and 
Fig.  (6)  is  a  sectional  projection  of  the  whole  train,  taken 


TOOTHED   GEARING.  197 

from  Fig.  (a),  on  the  line  AB,  The  double  curved  arms 
of  the  large  ($6")  gear  were  drawn  by  the  method  ex- 
plained in  §  XIV.,  under  Fig.  87.  It  may  be  remarked 
here  that  very  often,  perhaps  in  the  majority  of  cases, 
in  order  to  save  calculation  and  work,  the  pitches  for  all 
the  gears  of  a  train  are  taken  the  same.  Obviously, 
when  such  is  the  case,  the  common  pitch  must  be  taken 
equal  to  that  of  the  gear  which  transmits  the  greatest 
force  ;  in  the  last  example,  that  of  the  gear  Rf  (or  r' 
which  transmits  the  same  force).  Suppose  the  driving- 
gear  r  to  make  120  revolutions  per  minute  ;  then,  from 
formula  (7),  the  number  of  revolutions  per  minute  made 
by  R  is  120  X  Jf  —  53^.  The  gear  r'y  being  fixed  to 
the  same  shaft,  makes  the  same  number  of  revolutions 
as  R  ;  and  the  number  of  revolutions  per  minute  of  Rf, 
and  consequently  of  the  drum  r" ,  is  53^  X  \\  —  11.43. 
The  diameter  of  the  drum  is  14",  and  its  circumference 
43.98" :  hence  the  circumferential  velocity  of  the  drum, 
or  the  velocity  with  which  the  weight  will  be  lifted,  is 
43.98  X  H.43  _  4,i8g  feet  per  minute 

§  XVII.  —  Special  Applications  of  the  Principles  of  Toothed  Gearing. 

In  the  foregoing  pages  the  subject  of  toothed  gearing 
has  been  treated  in  so  far  as  it  relates  to  ordinary  ma- 
chinery only.  The  simple,  uniform,  rotary  motion  of 
the  spur  wheel,  bevel,  or  screw  gear,  the  continuous 
rectilinear  movement  of  the  rack  —  these  are  met  with 
daily  in  almost  every  shop  and  factory.  But  there  are 
many  special  cases  in  which  these  simple,  uniform  mo- 
tions are  not  sufficient.  According  to  the  work  which 
is  to  be  performed,  we  need,  in  one  case,  an  intermittent 


198 


TOOTHED   GEARING. 


rotary  or  rectilinear  motion  ;  in  another,  a  gradually  in- 
creasing or  decreasing  speed ;  and,  in  another,  a  recip- 
rocating movement.  These  variations  must  be  obtained 
from  the  uniformly  rotating  shop-shaft ;  and  there  arc 
few  fields  in  which  the  ingenuity  of  man  has  had  wider 
scope,  or  produced  more  variety  and  beauty  of  mechan- 
ism, than  in  that  of  special  gear-contrivances.  Some  of 
the  more  useful  and  common  of  these  many  special 
mechanisms  will  be  found  explained  in  the  following 
pages. 

Fig.99 


(i)  Spur  Gearing.  — Fig.  99  represents  a  pair  of 
"square"  or  "rectangular"  gears,  the  object  of  which 
is  to  obtain  a  varying  speed  for  the  driven  gear  <f  from 
the  uniform  rotary  motion  of  the  driver  c.  As  explained 
in  §  XL,  we  have  the  expression, 


n       R  .      Rn 

—,  =  TT,    or     n  =  -&-, 


in  which  R  and  n  are  the  radius  and  number  of  revolu- 
tions of  the  driver,  and  Rr  and  ;/  the  same  for  the 
driven  gear.  From  this  last  expression  it  is  plain,  that, 
if  we  increase  R ',  we  decrease  the  value  of  ;/;  if  we 


TOO  Til  ED    GEA  RING. 


199 


decrease  Rf,  we  increase  ;/;  if  we  increase  R,  we  in- 
crease ;/;  and,  if  we  decrease  R,  we  decrease  .«'...  In 
Fig.  99,  while  the  gears  are  in  the  position  shown,  the 
greatest  radius  of  the  driver  gears  with  the  smallest; 
radius  of  the  driven  gear :  the  speed  of  the  driven  gear 
is,  therefore,  then  at  its  maximum.  As  the  gears  revolve 
in  the  directions  indicated  by  the  arrows,  the  radius  of 
the  driver  gradually  decreases,  and  that  of  the  driven 
gear  gradually  increases,  until  the  points  x  and  a'  are  in 
contact.  The  speed  of  the  driven  gear,  therefore,  grad- 


Fig.lOO 


ually  decreases  during  this  eighth  of  a  revolution. 
From  the  moment  of  contact  between  the  points  x  and 
a',  the  reverse  action  takes  place,  and  the  speed  of  the 
driven  gear  gradually  increases  until  the  points  b  and  k 
are  in  contact.  Thus,  during  the  entire  revolution,  the 
driven  gear  continues  to  alternate  from  a  gradually 
decreasing  to  a  gradually  increasing  speed,  and  vice 
versa.  In  order  that  rectangular  gears  shall  work 
properly  together,  it  is  necessary,  first,  that  the  pitch 
peripheries  of  the  two  gears  be  equal  in  length,  and, 
second,  that  the  sum  of  the  radii  of  each  pair  of  points 


200  TOO  7^1  ED   GEARING. 

(points  which  come  into  contact  with  each  other)  on  the 
two  pitch  peripheries  be  equal  to  the  distance  between 
the  centres  of  the  gears.  These  two  conditions  sug- 
gest the  method  for  finding  the  pitch  periphery  (or 
pitch  line)  of  a  rectangular  gear  which  shall  properly 
gear  with  a  given  driver,  shown  in  Fig.  100,  which  is  as 
follows.  O  is  the  given  driver.  Since  the  smallest 
radius  of  the  driver  gears  with  the  greatest  radius  of 
the  driven  gear,  and  the  sum  of  these  two  radii  is  equal 
to  the  distance  between  the  centres,  make  aOr  =  R,  and 
O'  is  the  centre  for  the  required  gear.  Divide  the  peri- 
phery of  O  into  a  number  of  small  parts,  ai,  12,  23,  etc. ; 
and  from  the  point  O  as  a  centre,  and  Oi,  6>2,  O$,  etc., 
as  radii,  strike  arcs  cutting  the  line  of  centres  in  the 
points  b,  c,  d,  etc.  With  the  centre  O'  and  radii  Ofb,  O'c, 
O'd,  etc.,  strike  circle  arcs,  and  lay  off  the  arcs  ax,  xy, 
yz,  etc.,  equal  respectively  to  ai,  12,  23,  etc.,  taking 
care  that  the  points  x,  y,  z.,  etc.,  fall  upon  the  corre- 
sponding arcs,  of  which  the  point  O'  is  the  centre ;  so 
on,  until  the  entire  required  pitch  line  is  determined 
by  the  points  thus  found.  As  may  be  at  once  seen  by 
comparing  Figs.  99  and  100,  the  shape  of  the  driven 
periphery  depends  upon  the  amount  of  curvature  of  the 
"  corners  "  of  the  driver.  Thus,  for  very  slightly  curved 
corners,  the  driven  periphery  becomes  more  nearly  star 
shaped,  as  in  Fig.  100.  If  we  take  the  radius  of  curva- 
ture for  the  corners  (b'd ',  Fig. 99 )  equal  to  b'c,  the  gear 
peripheries  become  equal  and  similar,  and  the  gears 
square,  with  rounded  corners. 

Fig.  101  represents  a  pair  of  "triangular"  gears,  the 
object  of  which  is  to  obtain  an  alternating,  varying 
speed  from  the  uniformly  rotating  driver,  as  in  rec- 


TOOTHED    GEARING. 


20  T 


tangular  gears.  Triangular  gears  give  fewer  changes 
of  speed  per  revolution  than  rectangular  gears.  In 
Fig.  101,  C  being  the  driver  and  C  the  driven  gear,  the 
speed  of  the  latter  is  at  its  minimum  when  the  gears 
are  in  the  positions  shown  in  the  figure.  Since,  from 
these  positions,  the  radius  of  the  driver  gradually  in- 
creases, and  that  of  the  driven  gear  decreases,  as  far 
as  the  points  b  and  br,  the  speed  of  the  driven  gear 
will  gradually  increase  until  the  points  b  and  b'  are  in 
contact,  or  for  one-sixth  of  an  entire  revolution.  The 


reverse  action  will  then  take  place  until  the  points  c 
and  c'  are  in  contact,  and  so  on.  Thus  while,  in  rec- 
tangular gears,  each  gradually  increasing  or  decreasing 
period  takes  place  during  one-eighth  of  a  revolution,  in 
triangular  gears  each  of  these  periods  occupies  one- 
sixth  of  a  revolution  ;  that  is,  in  rectangular  gears  there 
are  eight  alternately  increasing  and  decreasing  periods 
in  one  entire  revolution  of  the  driven  gear,  and  in  trian- 
gular gearing  there  are  but  six. 

In  "elliptical"  gears  (shown  in  Fig.  102)  we  have 
still  another  means  of  obtaining  the  same  result,  with 
the  difference,  that,  in  elliptical  gears,  each  period  of 


202 


TOOTHED   GEARING. 


gradually  increasing  and  decreasing  speed  takes  place 
during  one-fourth  of  a  revolution :  in  other  words, 
there  are  but  four  periods  of  increasing  and  decreasing 

Fig.  102 


speed  during  one  entire  revolution  of  the  driver.  To 
construct  the  pitch  lines  of  triangular  and  elliptical 
gears,  we  proceed  as  already  explained,  under  Fig.  100, 
for  rectangular  gears.  Fig.  103  represents  a  pair  of 

Fig.  103 


"scroll"  gears ;  c  being  the  driver,  and  c  the  driven 
gear.  From  the  positions  shown  in  the  figure  (in  which 
the  greatest  radius  of  the  driver  gears  with  the  smallest 
radius  of  the  driven  gear),  as  the  gears  revolve  in  the 


TOOTHED   GEARING. 


203 


directions  indicated  by  the  arrows,  the  radius  of  the 
driver  gradually  and  uniformly  decreases,  while  that  of 
the  driven  gear  gradually  and  uniformly  increases.  The 
speed  of  the  driven  gear  is  therefore  at  its  maximum 
when  the  gears  are  in  the  positions  shown,  and  gradu- 
ally and  uniformly  decreases  during  the  entire  revolu- 
tion. The  moment  before  the  positions  shown  in  the 
figure  are  reached,  the  smallest  radius  of  the  driver 
gears  with  the  greatest  radius  of  the  driven  gear :  the 
speed  of  the  latter  is  then  at  its  minimum,  and  sud- 
denly (as  the  gears  assume  the  positions  in  the  figure) 


Fig, 104 


changes  to  its  maximum.  To  construct  the  pitch  lines 
for  a  pair  of  scroll  gears,  proceed  as  follows.  Con- 
struct the  square  1234  (Fig.  104),  each  side  of  which 
is  equal  to  one-fourth  the  distance  <?/,  which  determines 
the  rapidity  of  variation  in  the  speed  of  the  driven  gear. 
Produce  the  sides  of  the  square,  as  shown  in  the  figure. 
From  the  point  I  as  a  centre,  and  a  radius  \a,  strike 
the  arc  ab ;  with  the  point  2  as  a  centre,  and  2b  as  a 
radius,  strike  the  arc  be;  with  centre  3,  and  radius  3^, 
strike  the  arc  cd ;  and  with  centre  4,  and  radius  4</, 
strike  the  arc  df.  These  four  arcs  together  form  the 
pitch  line  of  the  driver,  the  axis  of  which  is  at  the  cen- 


204 


TOOTHED   GEARING. 


tre  x  of  the  square  1234.  Make  aC'=fx,  and  Cr  is 
the  centre  for  the  driven  pitch  line ;  after  which  pro- 
ceed to  find  points,  and  construct  the  pitch  line  ab'c'd' 
as  explained,  for  rectangular  gears,  under  Fig.  100. 

The  mechanism  represented  in  Fig.  105  is  known  as 
"  sector  "  gears,  and  the  object  is  to  obtain  a  series  of 


Fig.  105 


different  uniform  speeds.  In  the  figure,  C  is  the  driver, 
and  C'  the  driven  gear.  As  long  as  the  arcs  ab  and  ab' 
are  in  gear,  the  speed  of  the  driven  gear  is  the  same. 
When  the  arcs  cd  and  cd'  come  into  gear,  the  speed  of 
the  driven  gear  becomes  slower,  but  remains  the  same 
throughout  the  gearing  of  these  two  arcs.  Similarly, 
when  the  arcs  ef  and  e'f  come  into  gear,  the  speed  of 
the  driven  gear  becomes  still  slower,  but  uniform  during 
the  gearing  of  these  arcs.  Thus,  during  each  revolu- 
tion, the  driven  gear  has  three  periods  of  uniform  speed, 
each  differing  from  the  others.  In  order  that  sector 
gears  shall  work  properly  together,  it  is  necessary  that 
the  arcs  which  gear  together  be  equal  in  length  (ab  =  ab', 
cd  —  c'd',  etc.),  and  that  the  sum  of  the  arc  lengths  upon 
one  gear  be  equal  to  the  sum  of  the  arc  lengths  upon  the 
other  (ab  +  cd+ef  =  ab'+  c'd'+  e'f).  Also  the  sum  of 


TOOTHED   GEARING.  205 

the  radii  of  each  two  arcs  which  gear  together  must  be 
equal  to  the  distance  between  the  centres  of  the  gears. 
Sector  gears  are  somewhat  difficult  to  construct,  be- 
cause considerable  care  must  be  taken  that  no  two 
sectors  of  the  driver  gear  at  the  same  time  with  the 
driven  gear.  To  illustrate,  suppose  (Fig.  105)  that  the 
arcs  ab  and  ab'  gear  together  at  the  same  time  as  do 
the  arcs  ef  and  e'f'\  that  is,  that  the  last  few  teeth  of 
ab  gear  with  the  driven  gear  at  the  moment  when  the 
first  few  teeth  of  efdo  the  same.  The  driver  will  then 
strive  to  drive  the  driven  gear  at  its  maximum  and 
minimum  speeds  at  the  same  time,  — an  attempt  which 
must  obviously  result  in  a  fracture.  In  the  figure,  the 
arc  </ ceases  to  gear  with  the. driven  gear  at  the  mo- 
ment when  the  arc  ab  begins  to  gear.  Thus  each  arc 
of  the  driver  must  escape  gear  just  in  time  for  its  suc- 
cessor to  begin  gear,  and  yet  leave  between  these  events 
no  appreciable  interval  to  disturb  the  uniformity  of  mo- 
tion. Fig.  106  represents  a  peculiar  kind  of  spur  wheel 
and  pinion.  The  wheel  has  two  sets  of  teeth,  one  set 
being  on  each  side  ;  and  the  teeth  of  the  two  sets  alter- 
nating in  position,  as  shown  in  the  figure.  The  pinion 
consists  of  two  heart-cams,  so  arranged  that  each  gears, 
in  turn,  with  one  set  of  teeth  of  the  spur  wheel.  By 
this  means  a  very  slow  motion  is  obtained  for  the  spur 
wheel,  which  is  moved  through  a  distance  of  two  teeth 
at  each  revolution  of  the  cam-pinion.  In  the  figure,  the 
cam  a  leaves  the  tooth  a'  some  time  before  the  cam  /; 
comes  into  contact  with  the  tooth  bf:  during  this  time, 
therefore,  the  spur  wheel  remains  motionless,  or,  in 
other  words,  the  motion  of  the  spur  wheel  is  intermit- 
tent. The  length  of  time  during  which  the  driven  gear 


2O6 


TOOTHED   GEARING. 


remains  motionless  depends  upon  the  shape  of  the  cams. 
Thus,  if  we  give  to  the  cam  b  the  shape  indicated  by 
the  dotted  outline,  the  cam  will  engage  sooner  with  the 
tooth  //:  consequently  the  period  of  rest  will  be  shorter, 
and  the  period  of  motion  longer.  Also,  if  the  cams 
differ  from  each  other  in  shape,  the  periods  of  rest  pre- 


Fig.  106 


Fig. 107 


ceding  the  engagement  of  the  two  cams  will  be  of 
different  lengths,  and  the  motion  of  the  driven  gear 
will  be  rendered  still  more  variable.  Fig.  107  repre- 
sents the  device  known  as  "  stepped  "  gears.  This  ar- 
rangement is  used  when  very  heavy  powers  are  to  be 
transmitted,  and  is  met  with  sometimes  in  large  and 
powerful  machine  tools.  In  the  figure,  each  of  the 
shafts  c  and  /  bears  three  spur  wheels  ;  the  pitches, 
diameters,  etc.,  being  equal,  and  the  three  gears  being 
keyed  firmly  to  the  shaft.  The  gears  are  so  fixed  upon 
the  shaft  that  their  teeth  are  arranged  in  steps  along  the 
combined  face,  as  shown  in  the  figure  ;  i.e.,  each  gear  is 
turned  round  upon  the  shaft  slightly  farther  than  the 


TOOTHED    CEARfArG. 


207 


Fig.  (08 


preceding  one,  so  that  instead  of  there  being,  say,  \\ 
teeth  gearing  with  the  driver  at  one  time,  as  is  the  case 
in  a  pair  of  ordinary  spur  wheels,  there  are  3  X  i \  =  4j. 
The  strain  is  thus  divided  among  three  gears,  and 
the  contrivance  is  capable  of  transmitting  three  times 
the  power  which  can  be  transmitted  by  one  pair  of  the 
gears.  The  device  represented  in  Fig.  108  consists  of  a 
mutilated  spur  driver  c,  a  spur  pinion  /,  and  a  mutilated 
internal  gear  d;  the  gears  c  and  d  are  fixed  upon  the 
same  shaft.  The  mutilated 
spur  wheel  c  drives  the  pinion 
c  in  the  direction  shown  by 
the  arrow,  until  the  point  a 
is  reached,  when  the  gear  c 
ceases  to  be  the  driver,  and 
the  mutilated  internal  gear 
takes  its  place.  This  drives 
the  pinion  c  in  a  contrary 
direction,  until  the  point  k 
is  reached,  when  c  again  be- 
comes the  driver,  and  again 
reverses  the  direction  of  rotation  of  the  pinion.  The 
gear  c  being  small  in  diameter,  and  the  gear  d  large, 
the  former  drives  the  pinion  c'  at  a  slow  speed,  and  the 
latter  gives  to  it  a  high  speed.  The  mechanism  is  there- 
fore useful  where  a  slow  forward  motion  and  a  quick 
return  are  needed,  as  in  the  planer,  and  other  machine 
tools. 

The  arrangement  represented  in  Fig.  109,  which  con- 
sists of  two  spur  wheels  and  a  mutilated  spur  driver,  is 
intended  to  give  to  the  spur  wheels  c'  and  d  an  alter- 
nating, intermittent  motion.  The  driver  c,  rotating  in 


208 


TOOTHED   GEARING. 


Fig.109 


the  direction  indicated  by  the  arrow,  drives  the  spur 
wheel  c  in  the  direction  shown,  until  the  tooth  x  comes 
into  gear.  From  that  moment  the  driver  acts  upon  the 
spur-wheel  d,  which  it  drives  in  the  same 
direction  as  that  given  to  the  gear  d ' , 
When  the  tooth  x  comes  into  contact 
with  the  gear  d,  the  driver  ceases  to  act 
upon  this  gear,  and  returns  to  the  gear  /. 
Thus  the  intermittent  motions  of  the 
two  spur  wheels  are  made  to  alternate ; 
the  gear  cf  remaining  at  rest  while  the 
gear  d  is  in  motion,  and  contrariwise. 

In  Fig.  i  JO  the  mutilated  driving  pin- 
ion engages  alternately  with  the  racks  / 
and  d,  which  it  drives,  at  the  same  speed, 
in  opposite  directions.  The  two  racks 
being  rigidly  fixed  to  one  frame,  a  recip- 
rocating rectilinear  motion  is  given  to 
the  frame ;  the  forward  and  return  mo- 
tions being  the  same  in  velocity. 

If  the  two  racks  are  mutilated  and  the  pinion  entire, 
as  in  Fig.  1 1 1,  the  mutilations  being  alternately  situated 
Fig.no  on  the  two  racks,  a  con- 

^      tinuous  rectilinear  mo- 
£&  ^\\  tion  of  the  rack  frame 
\A  will  give  to  the  pinion 
J J  an    alternating    rotary 
yj  motion ;  the   speeds  of 
S     advance  and  return  mo- 


tion being  the  same.  If  the  rack  mutilations  are  of 
different  lengths,  the  motion  of  the  pinion  will  be  vari- 
able ;  the  pinion  moving  over  a  greater  distance  when 


TOOTHED   GEARING.  2OQ 

engaging  with  a  long  toothed  part,  and  a  less  distance 
when  engaging  with  a  part  of  the  rack  containing  a  few 
teeth  only.  Fig.  1 12  represents  a  device  for  obtaining  a 
uniform  rectilinear  motion  in  one  direction,  and  a  sudden 
return  motion.  The  mutilated  pinion,  rotating  continu- 
ously in  the  direction  shown  by  the  arrow,  imparts  a 
downward  motion  to  the  rack  until  the  toothless  part  of 
the  pinion  is  reached.  The  rack,  being  then  free,  is  lifted 
quickly  into  its  original  position  by  means  of  the  weight 
Wj  cord  and  pulley  K.  This  arrangement  is  sometimes 
used  on  special  auto-mat-  Fig. in 

ic  drills,  in  which  case 
the  rack  is  fixed  upon  a 
frame  within  which  the 
drill  spindle  works.  The 
spindle  bears  a  raised 
ring,  which  fits  into  an 
annular  depression  within  the  frame.  This  allows  the 
spindle  to  revolve  freely,  still  enables  the  pinion  and 
rack  to  give  to  the  spindle  the  rectilinear  motion  neces- 
sary for  the  feed ;  and  at  the  proper  time  the  weight 
returns  the  spindle  to  its  original  position  in  readiness 
to  repeat  the  desired  operation. 

In  Fig.  113  the  mutilated  driver  C  acts  upon  the 
gear  C',  driving  it  uniformly  in  the  direction  shown  by 
the  arrow,  until  the  toothless  parts  are  opposite  each 
other,  when,  the  gear  C'  being  free,  the  weight  W  falls, 
and  quickly  carries  the  gear  C'  into  such  a  position  that 
the  driver  again  gears  with  it ;  and  the  same  action 
again  takes  place.  Thus  a  variable  rotary  motion  is 
imparted  to  the  driven  gear,  —  slow  when  the  driver 
acts  upon  it,  and  fast  when  it  is  acted  upon  by  the 


/A/WV\J 


2IO 


TOOTHED   GEARING. 


weight  W.  If  we  change  the  numbers  of  teeth,  so  that, 
when  the  teeth  of  the  driver  and  driven  gear  cease  con- 
tact, the  weight  has  the  position  W  (the  directions  of 
rotation  being  the  same  as  in  the  figure),  the  weight,  in 
falling,  will  carry  the  gear  C'  in  a  direction  contrary  to 
that  imparted  by  the  driver,  and  the  motion  of  the  driven 
gear  will  be  an  alternating  or  oscillating  one,  made  up 
of  a  slow  forward  and  a  quick  return  movement. 


Fig. 1 12 


Fig. 113 


(2)  Bevel  Gearing.  —  Under  the  head  of  special  appli- 
cations of  bevel  gearing  we  propose  to  include  some 
pairs  of  gears  which  are  not,  strictly  speaking,  bevel 
gears,  since  the  teeth  are  not  "bevelled,"  but  which 
resemble  bevels  in  that  their  shafts  are  not  parallel,  but 
form  either  oblique  or  right  angles  with  each  other. 
An  example  of  such  a  pair  is  seen  in  Fig.  114.  The 
gear  C  is  an  ordinary  spur  wheel,  and  ^'what  is  termed 
a  "crown  gear."  The  teeth  of  the  latter  gear  are  made 
so  thin  that  their  sides  are  practically  parallel,  and 


TOOTHED    GEARING. 


21  i 


Fig.ii4 


hence  gear  with  the  spur  wheel,  notwithstanding  the 

fact  that  their  side  lines  all  intersect  at  the  centre  of 

the  pitch  circle.     Because 

of  the  necessarily  thin  teeth 

of  the  crown  gear,  such  a 

pair   as    is    shown    in    the 

figure  can  be  used  only  for 

the   transmission    of   very 

slight   powers.     They  are 

very  seldom  seen  in  prac- 

tice,    except    in     models, 

mathematical  instruments, 

and  such  like. 

Fig.  115  represents  a 
crown  gear,  C',  which  en- 
gages with  a  wide-faced  spur  driver,  C.  The  shaft  of  the 
crown  gear  is  set  eccentrically,  instead  of  in  the  centre 
of  the  gear  :  hence  a  variable  motion  is  given  to  it  by 


Fig, 115 


the  uniform  rotary  motion  of  the  driver.  The  motion 
of  the  crown  gear  is  fast  when  the  gears  are  in  the  posi- 
tions shown  in  the  figure,  and  gradually  decreases  until 
the  largest  radius  comes  into  gear,  when  the  reverse 


212 


TOOTHED   GEARING. 


action  takes  place  for  the  remaining  half-revolution. 
The  face  width,  ab,  of  the  driver,  must  be  at  least  equal 
to  the  difference  between  the  greatest  and  smallest 
inner  radii  of  the  crown  gear  plus  the  thickness  of  the 
crown  gear  teeth,  xy. 

Fig.  1 1 6  represents  a  contrivance  for  obtaining  three 
different  uniform  rotary  motions  for  the  shaft  C'  from 
a  uniformly  revolving  shaft  C,  the  two  shafts  being  at 


Fig. 116 


Fig.117 


right  angles  with  each  other.  The  wheel  C  has  three 
sets  of  projecting  pins,  arranged  in  circles  of  different 
diameters,  as  shown  in  the  figure.  The  pitches  (dis- 
tance between  the  centres  of  two  adjacent  pins)  of  all 
the  circles  are  equal.  The  gear  C'  has  a  slotted  face, 
the  slots  being  slightly  larger  than  the  pins  of  the  wheel 
C,  and  equally  distant  from  each  other.  By  sliding  the 
gear  C'  along  its  shaft,  it  may  be  made  to  engage  at 
will  with  either  of  the  three  circles  of  the  wheel  C,  thus 
obtaining  a  quick  or  slow  motion  as  may  be  required. 


TOOTHED   GEARING. 


213 


Fig.  117,  which  represents  a  device  for  giving  two 
different  velocities  to  the  same  shaft,  consists  of  a 
driving  bevel  c>  and  two  driven  bevels  c  and  d,  of  differ- 
ent diameters,  and  running  on  the  same  shaft.  The 
bevel  d,  being  smaller  than  the  bevel  c',  is  driven  at  a 
greater  speed,  and  in  a  direction  contrary  to  that  of  /. 
The  bevel  c  is  fixed  to  a  collar  or  hollow  shaft,  g,  which 
fits  over  the  shaft  k,  thus  allowing  it  to  revolve  in  a  con- 
trary direction.  If  the  driving  bevel  c  is  mutilated,  and 


Fig. 118 


the  bevels  c'  and  d  fixed  to  the  shaft  k,  an  alternating 
rotary  motion  will  be  given  to  the  shaft,  the  alternations 
being  at  different  speeds.  The  same  result  may  be 
obtained  for  the  shaft  c  by  mutilating  the  gears  c'  and  d 
so  that  the  toothed  part  of  one  is  opposite  the  toothless 
part  of  the  other,  and  making  the  bevel  c  the  driven 
gear. 

Fig.  118  represents  a  method  of  obtaining  an  alter* 
nating  rotary  motion  from  a  uniformly  rotating  shaft, 
the  driving  and  driven  shafts  being  at  right  angles  with 


214 


TOOTHED   GEARING. 


Fig.  119 


each  other.  The  mutilated  driving  bevel  c  drives  the 
shaft  c'd  alternately  in  opposite  directions,  according  as 
it  gears  with  the  bevel  c'  or  d.  The  speeds  of  the  for- 
ward and  return  motions  are  the  same,  since  the  bevels 
/and  d  are  of  the  same  diameter.  This  contrivance 
was  once  used  to  give  the  reciprocating  motion  to 

planer-beds;  a  thread  on  the 
shaft  c'd,  which  worked  in  a 
female  thread  in  the  bed,  pro- 
ducing the  rectilinear  motion. 
The  arrangement  soon  fell 
into  disuse,  for  the  reason  that 
as  much  time  was  required  for 
the  return  as  for  the  forward 
motion,  a  waste  which  is  now 
obviated  by  the  more  modern 
"quick  return." 

The  device  represented  in 
Fig.  119  is  intended  to  trans- 
mit a  gradually  increasing  speed  to  a  shaft  from  the 
uniform  rotary  motion  of  a  shaft  at  right  or  oblique 
angles.  The  scroll  bevel  C  is  the  driver,  and  the  ordi- 
nary bevel  C  the  driven  gear.  Starting  with  the  small- 
est radius  of  the  scroll  bevel  (at  the  point  a)  in  gear  with 
the  driven  bevel,  and  rotating  in  the  direction  indicated 
by  the  arrow,  the  radius  gradually  and  steadily  increases 
until  the  bevels  assume  the  positions  shown  in  the 
figure  :  consequently  the  speed  of  the  driven  bevel 
gradually  and  steadily  increases  during  the  entire  revo- 
lution. The  toothed  part  of  the  scroll  bevel  may  be 
carried  farther  than  in  the  figure,  as  indicated  by  the 
dotted  lines,  and  the  described  action  thus  made  to  take 


TOOTHED   GEARING. 


215 


place  during  more  than  one  revolution.  The  shaft  of 
the  driven  bevel  carries  a  feather,  which  allows  the 
bevel  to  slide  along  it  without  interfering  with  the  ro- 
tary motion.  In  the  figure,  when  the  described  action 
begins,  the  driven  bevel  C  is  in  its  highest  position  on 
the  shaft ;  and,  as  during  the  rotation  the  radius  of  the 
driving  bevel  increases,  the  former  bevel  is  forced  down- 
ward upon  its  shaft  until  the  positions  shown  in  the 
figure  are  reached.  If  the  scroll  bevel  be  made  to  rotate 
in  a  direction  opposite  to  that  indicated  in  the  figure, 
it  is  plain  that  the  teeth,  not  being  prevented  from  so 


Fig.120 


doing  by  the  converging  of  their  lines,  will  lift  out  of 
gear  as  the  radius  decreases,  and  thus  destroy  the 
action. 

Fig.  1 20  represents  a  peculiar  kind  of  bevel  gear, 
more  properly  a  pair  of  right-angle  gears.  The  driving 
gear  C  bears  upon  its  circumference  small  rollers,  which 
gear  into  curved  projections,  or  grooves,  in  the  face  of 
the  driven  gear  C't  and,  by  rolling  down  these  curves, 
give  to  the  driven  gear  a  rotary  motion  at  right  angles 
with  that  of  the  driver.  The  motion  of  the  driven  gear 
depends  upon  the  shape  of  the  projections.  If  these 
are  curved,  the  curves  being  more  oblique  to  the  verti' 


216 


TOOTHED   GEARING. 


cal  at  the  bottoms  than  at  the  tops,  as  in  the  figure,  the 
motion  of  the  driven  gear  will  be  variable,  —  slow  when 
each  roller  of  the  driver  gears  with  the  upper  part  of  a 
projection,  and  gradually  faster  as  the  roller  progresses 
downward.  If,  instead  of  being  curved,  the  profiles  of 
the  projections  are  straight  lines,  the  motion  of  the 
driven  gear  will  be  nearly  uniform. 

The  motions  described  under  Fig.  113  may  be  trans- 
mitted from  one  shaft  to  another  at  right  or  oblique 

Fig.  121 


angles,  by  using  mutilated  bevels  in  place  of  the  spur 
gears  shown  in  that  figure.  Fig.  121  represents  an  ar- 
rangement of  bevels  known  as  the  " mangle  wheel"  and 
pinion,  the  object  of  which  is  to  obtain  an  alternating 
rotary  motion  for  the  mangle  wheel  Cf.  This  wheel 
has  teeth  upon  both  sides,  one  side  only  being  shown  in 
the  figure.  As  the  driving  bevel  C  rotates,  it  drives  the 
mangle  wheel  in  the  direction  indicated  by  the  arrow, 
until  the  opening  ef  is  reached.  At  this  point  the  guide 
a  comes  into  contact  with  the  shaft  of  the  driver,  which 


TOOTHED   GEARING. 

it  forces  downward  through  the  opening,  and  into  such 
a  position,  that  the  driver  gears  with  the  teeth  on  the 
other  side  of  the  mangle  wheel.  The  latter  is  then 
driven  in  an  opposite  direction,  until  the  opening  cf  is 
again  reached,  when  the  guide  b  lifts  the  driver  up 
through  the  opening  into  gear  with  the  first-mentioned 
side  of  the  mangle  wheel.  This  operation  is  repeated 
indefinitely;  the  mangle  wheel  making  one  entire  revo- 
lution alternately  in  each  direction.  The  shaft  of  the 
driving  bevel  carries  a  universal  joint,  x,  which  allows 

Fig. 122 


it  enough  freedom  of  motion  to  fall  and  rise  through 
the  opening  in  the  mangle  wheel. 

Fig.  122  represents  an  arrangement  of  bevel  gears, 
the  object  of  which  is  to  produce  a  double  or  half 
speed  ;  the  three  bevel  gears  having  the  same  diameter. 
The  bevel  c  is  rigidly  fixed  (so  that  it  cannot  rotate)  to 
the  bed  of  the  mechanism,  and  the  shaft  ab  runs  loosely 
through  it.  The  bevel  cr  runs  loose  upon  the  shaft  ab, 
which  carries  a  short,  right-angle  shaft,  cf.  Upon  this 
right-angle  shaft  the  bevel  d  runs  loose.  If,  now,  a 
rotary  motion  be  given  to  the  shaft  ab,  the  right-angle 


2l8  TOOTHED   GEARING. 

shaft  ef,  and  with  it  the  bevel  d,  will  be  made  to  revolve 
in  a  vertical  plane  about  the  axis  ab.  The  bevel  d  will 
also,  by  its  gearing  with  the  fixed  bevel  c,  be  made  to 
rotate  upon  its  own  axis,  ef.  Since  the  bevels  c  and  d 
are  of  the  same  diameter,  the  speeds  of  these  two  rota- 
tions will  be  the  same  :  therefore  the  bevel  d  will  trans- 
mit to  the  bevel  c'  the  effect  of  two  speeds,  each  equal 
to  that  of  the  shaft  ab.  And,  since  the  speeds  are  in 
the  same  direction,  the  bevel  c'  will  be  made  to  rotate 
about  the  shaft  ab  with  a  speed  equal  to  twice  that  of 
the  shaft  ;  that  is,  while  the  shaft  ab  makes  one  entire 
revolution  in  a  given  direction,  the  bevel  c'  will  make 
two  revolutions  in  the  same  direction.  If  the  bevel  c' 
be  made  the  driver,  its  rotary  motion  will  transmit  to 
the  bevel  d  a  rotary  motion  about  its  axis  ef,  and,  by 
means  of  the  fixed  bevel  c,  also  a  revolving  motion  in 
a  vertical  plane  about  the  axis  ab.  The  bevels  having 
equal  diameters,  half  the  speed  of  the  driver  is  trans- 
mitted in  the  rotation  of  the  bevel  d  about  its  axis  ef, 
and  half  in  the  vertical  rotation  about  the  axis  ab  ;  that 
is,  the  shaft  ab  will  be  made  to  rotate  with  a  speed 
equal  to  one-half  that  of  the  driver  :  while  the  driver  c' 
makes  two  entire  revolutions  in  a  given  direction,  the 
shaft  ab  will  make  one  revolution  in  the  same  direction. 
The  relative  speeds  of  the  shaft  ab  and  the  bevel  c  may 
be  varied  by  changing  the  relative  diameters  of  the 
bevels. 

(3)  Screw  Gearing.  —  Fig.  123  represents  a  very  com- 
mon mode  of  transforming  uniform  rotary  into  uniform 
rectilinear  motion.  The  threaded  shaft  ab,  rotating 
upon  its  axis,  and  restrained  from  other  motion  by  the 
collars  xy  and  fg,  works  in  a  female  thread  in  the  piece 


TOOTHED   GEARING. 


Fig. (23 


r,  thus  giving  to  the  latter  piece  a  rectilinear  motion 
upon  the  slides  k,  k.  By  reversing  the  direction  of  rota- 
tion of  the  shaft  ab,  the 
direction  of  the  motion  of 
the  piece  C  will  also  be 
changed.  This  device  is 
seen  in  the  leading-screws 
of  lathes,  in  the  arrange- 
ment for  feeding  the  tool 

holders    in    planing    ma-    

chines,  drills,  etc. 

In  Fig.  1 24  the  cylinder  C  has  right   and  left  spiral 
grooves  cut  in  its  surface,  as  shown  in  the  figure.     The 


Fig.i24 


Fig.  125 


tooth  k  of  the  slide  /  fits 
into  the  grooves.  Upon 
giving  to  the  cylinder  a 
rotary  motion  about  its  axis 
ab  (supposing  the  tooth  k  <• 
to  be  working  in  the  right- 
hand  groove),  the  slide  /  is  made  to  move  along  the 
frame  d,  upon  which  it  rests,  until  the  end  of  the  groove 
is  reached,  when  the  tooth  runs  into 
the  left-hand  groove,  and  the  slide  f 
returns  in  the  opposite  direction. 
Thus  a  reciprocating  rectilinear  mo- 
tion is  obtained  from  the  uniform 
rotary  motion  of  the  cylinder. 

In  Fig.  125  a  uniform  rotary  motion 
of  the  pulley  C  gives  to  the  slide  / 
a    reciprocating    rectilinear    motion 
along  the  frame  d,  by  means  of  the  zigzag  groove  upon 
the  pulley  surface,  in  which  the  tooth  k  of  the  slide 


220 


TOOTHED    GEARING. 


works.  By  giving  to  the  groove  in  the  surface  of  the 
pulley  the  proper  shape,  the  motion  of  the  slide  f  may 
be  made  uniform,  variable,  or  intermittent. 

Fig.  126  represents  a  device  for  transforming  uniform 
rotary  motion  into  two  rectilinear  motions  in  opposite 
Fjg.  126  directions.    The  shaft 

ab,  which  carries  the 
right  and  left  screw- 
threads  shown  in  the 
figure,  rotates  within 
its  bearing  O.  The 
right  and  left  screws 
work  in  female  screws  within  the  pieces  C  and  C' : 
consequently  these  pieces  are  driven  in  contrary  direc- 
tions, approaching  each  other,  or  receding  from  each 
other,  according  to  the  direction  of  rotation  of  the  shaft 

ab.  This  arrangement  is 
used  in  presses  of  .various 
kinds,  the  arms  indicated 
by  the  dotted  lines  being 
drawn  together  at  their 
tops  by  the  action  of  the 
screws,  and  the  point  x 
being  forced  slowly  down- 
ward with  great  force  and 
steadiness.  The  arrange- 
ment of  worm  wheels  rep- 
resented in  Fig.  127  is 
intended  to  produce  two  uniform  rotary  motions  in  oppo- 
site directions.  The  right  and  left  worms  on  the  shaft 
ab  cause  the  worm  wheels  C  and  C'  to  rotate  in  opposite 
directions  when  the  shaft  is  given  a  rotary  motion.  The 


Fig.  127 


TO  O  THE!)   GEA  R  L  \  TG. 


221 


Fig.  129 


same  effect  may  be  obtained  with  one  worm,  by  gearing 
with  it  two  worm  wheels,  C  and  d,  on  opposite  sides  of 
the  shaft,  as  indicated  by  the  dotted  circle. 

Fig.  128  represents  a  peculiar  example  of  screw  gear- 
ing.    The  disk  C  carries  upon  its  side  Fig.i28 
an   elevated   spiral,   as    shown   in  the 
figure.    This  spiral  gears  with  an  ordi- 
nary spur  gear  Cf,  the  shaft  of  which 
is  at  right  angles  with  that  of  the  disk. 
At  each  revolution  of  the  disk  C,  the 
constantly  changing  radius  of  the  spi- 
ral causes  the  spur  gear  to  rotate  for 
a  distance  equal  to  one  tooth  ;  the  pitch  /  of  the  spiral 
being  equal  to  the  pitch  of  the  spur  gear.     By  gearing 
with  the  spiral  two  spur  gears  (the 
second  is  indicated  in  the  figure  by 
the   dotted    lines),    motion    may  be 
transmitted  from  the  spiral  to  two 
shafts    at   right    angles    with    each 
other.     In  a  like  manner  the  spiral 
may  be  made  to  drive  several  spur 
gears   at   once,  the    shafts    making 
oblique  angles  with  each  other. 

In  Fig.  129  we  have  represented  a 
"  side  "  worm  wheel  C',  and  worm. 
The  former  carries  upon  its  side  pro- 
jections or  teeth,  as  shown  in  the  fig- 
ure ;  and  the  worm  on  the  shaft  abt 
gearing  with  these  teeth,  causes  the 
wheel  C'  to  rotate  uniformly,  the  ac- 
tion being  similar  to  that  of  an  ordinary  worm  and  wheel. 
By  gearing  with  the  worm  two  side  worm  wheels  (the 
second  being  indicated  in  the  figure  by  the  dotted  circle 


222 


TOOTHED   GEARING. 


d\  the  teeth  being  on  the  sides  of  the  wheels  which 
face  towards  each  other,  two  uniform  rotary  motions  in 
opposite  directions  may  be  obtained,  as  in  Fig.  127. 
The  motions  described  under  Fig.  113  may  be  obtained 
for  shafts  at  right  angles  with  each  other  by  substitut- 
ing for  the  mutilated  spur  gears  a  worm  and  mutilated 
worm  wheel. 

Fig.  130  represents  a  kind  of  worm  and  worm  wheel 
sometimes  used  to  transmit  very  heavy  powers.     The 
Pig.,30  primitive   surface  cdef,  of  the 

worm,  instead  of  being  a  right 
cylinder,  as  in  ordinary  worms, 
is  a  solid  of  revolution  gener- 
ated by  the  revolution  of  the 
circle  arc  cf  about  the  axis  ab. 
The  object  of  this  is  to  obtain 
a  contact  of  several  teeth  at 
one  time.  In  the  figure*  seven 
teeth  of  the  worm  are  in  gear 
at  the  same  time  with  the  teeth 
of  the  worm  wheel,  and  each 
tooth  sustains  an  equal  share 
of  the  transmitted  strain.  In  Fig.  127  only  two  teeth 
of  the  worm  are  in  gear  at  one  time  with  the  teeth  of  the 
driven  wheel  C.  If,  therefore,  we  have  to  transmit  such 
a  force  that  the  strain  on  the  teeth  is  10,000  pounds,  for 
example,  each  tooth  of  the  worm  in  Fig.  127  will  sustain 


a  strain  of  -i.Q-.2-0  _ 


pounds  ;  while  under  similar 


circumstances  each  tooth  of  the  worm  in  Fig.  130  will 
sustain  a  strain  of  -1M.Q.&  z=  1,430  pounds  :  in  other 
words,  the  latter  worm  is  capable  of  transmitting  |  =  3| 
times  the  force  of  the  former  worm  with  the  same  strain 
upon  each  tooth. 


APPENDIX. 


THE  present  tendency  among  mechanical  men  in  favor 
of  the  use  of  the  diametral  instead  of  the  older  and  more 
widely  known  circumferential  pitch,  together  with  the 
increasing  importance  of  cut  gears  (in  the  construction 
of  which  the  diametral  pitch  seems  to  be  especially  con- 
venient), has  induced  the  author  to  devote  an  appendix 
to  the  brief  discussion  of  the  relative  values  of  the  two 
kinds  of  pitch,  to  a  brief  explanation  of  the  method  of 
constructing  cut  gears,  and  to  the  working-out  of  simple 
rules  and  formulas,  by  means  of  which  all  the  necessary 
calculations  may  be  made  without  the  use  of  the  circum- 
ferential pitch.  From  §  X  we  have  the  expression 

pd  =  -,  in  which  pd  and  /  represent  respectively  the 

/ 

diametral  and  circumferential  pitch,  and  TT  the  irrational 
constant  3.14159-}-.  The  following  table  gives  values 
for  the  diametral  pitch,  for  different  circumferential 
pitches,  in  inches.  A  glance  at  the  table  will  show,  that, 
in  the  list  of  most  common  circumferential  pitches,  not 
one  corresponds  to  a  diametral  pitch  of  whole  numbers, 
or  even  exact  eighths,  sixteenths,  thirty-seconds,  etc. 
In  fact,  the  diametral  pitch  can  be  a  whole  number  only 

223 


224 


TOOTHED   GEARING. 


when  the  corresponding  circumferential  pitch  is  an  exact 
divisor  of  the  irrational  constant  TT,  a  condition  which 
is  not  at  all  likely  to  be  fulfilled. 


p 

A* 

P 

A* 

I 

25.1327 

3* 

0.8976 

I 

12.5664 

4 

0.7854 

i 

6.2832 

4* 

0.6981 

} 

4.1888 

5 

0.6283 

I 

3.1416 

5* 

0.5711 

II 

2.7926 

6 

0.5236 

I* 

2.5132 

6£ 

0.4833 

I» 

2.0944 

7 

0.4488 

If 

1.7952 

7* 

0.4188 

2 

1.5708 

8 

0.3927 

2* 

i  -3963 

9 

0.3491 

2£ 

1.2566 

10 

0.3142 

2| 

1.1424 

12 

0.2618 

3 

1.0472 

H 

0.2244 

For  this  reason,  in  all  gears  which  have  to  be  laid  out, 
—  as  cast  gears,  in  the  construction  of  which  the  pitch 
must  be  stepped  off  around  the  pitch  circumference  in 
the  drawings  and  pattern,  —  the  circumferential  pitch 
only  can  be  conveniently  used.  In  such  cases,  even  if 
we  have  given  the  diametral  pitch,  we  must  practically 
find  the  circumferential  pitch  before  we  can  properly 
divide  our  pitch  circumference,  and  lay  out  the  teeth. 
At  this  point  of  the  construction,  the  important  ques- 
tions are,  "  How  many  teeth  is  the  gear  to  have?"  and 
"  How  much  space  on  the  pitch  circle  does  each  tooth 
need?"  We  care  as  little  how  many  teeth  there  are  per 


TOOTHED   GEARfNG.  22$ 

inch  of  diameter  as  how  many  teeth  there  may  be  per 
pound  of  metal.  In  performing  the  calculations  neces- 
sary to  the  laying-out  of  gears,  the  diametral  pitch  offers 
no  advantages  over  the  circumferential.  Thus,  to  ob- 
tain the  number  of  teeth  with  the  latter  pitch,  we  divide 
the  pitch  circumference  (an  irrational  quantity)  by  the 
pitch  ;  while  in  using  the  former  pitch  the  case  is  no 
better,  for,  to  find  the  number  of  teeth  in  the  gear,  we 
must  multiply  the  pitch  diameter  by  the  diametral  pitch 
(itself  an  irrational  quantity).  Again  :  the  rules  and 
formulas  for  the  tooth  dimensions  at  present  in  use  in 
the  shops  are  in  terms  of  the  circumferential  pitch,  —  for 
example,  the  formulas  /=  2/,  or  /=  2\p,  h  =  o.?/,  etc., 
given  in  the  preceding  pages,  —  and,  while  using  the 
diametral  pitch,  we  must  either  obtain  the  circumferen- 
tial pitch  in  order  to  find  our  tooth  dimensions,  or  devise 
and  introduce  new  rules  and  formulas  in  terms  of  the 
diametral  pitch.  The  author  having  taken  the  pains  to 
ask  a  considerable  number  (68)  of  draughtsmen  and 
pattern-makers  in  the  States  of  Jtfew  York,  Pennsylvania, 
New  Jersey,  and  Connecticut,  their  preferences,  finds 
that  a  very  large  majority  (61  to  7)  of  those  spoken  to 
favor  the  use  of  the  old  circumferential  pitch.  This 
would  seem  to  indicate,  that,  while  the  same  theorists 
who  are  striving  to  force  upon  the  American  mechanic 
the  French  metric  system  are  clamoring  for  an  absolute 
discontinuance  of  the  use  of  the  old  pitch,  the  practical 
mechanic,  who  does  the  measuring  and  constructing, 
goes  steadily  on  with  his  work,  looking  neither  to  the 
right  for  a  "centimeter,"  nor  to  the  left  for  a  "diametral 
pitch." 

But  while,  according  to  the  opinion  and  experience  of 


226  TOOTHED   GEARING. 

the  author,  the  diametral  pitch  is  of  no  practical  use  in 
cast  gears,  it  cannot  be  reasonably  disputed,  that,  in  the 
construction  of  cut  gears,  this  pitch  has,  indeed,  advan- 
tages over  the  circumferential,  and  for  this  reason 
deserves  the  attention  and  respect  of  every  intelligent 
mechanic. 

In  the  construction  of  cut  gears  the  wheels  are  first 
cast  without  teeth,  the  entire  thickness  of  the  rim  being 
its  own  thickness  when  finished  plus  the  height  of  the 
teeth  (t  -f-  //).  The  spaces  between  the  teeth  are  then 
cut  out  by  means  of  revolving  circular  cutters,  the 
blades  of  the  cutters  being  as  nearly. as  possible  the 
shape  of  the  required  spaces.  In  order  to  properly  con- 
struct cut  gears,  a  shop  must  be  provided  with  different 
sets  of  cutters,  corresponding  to  the  different  pitches 
and  diameters  of  gears.  The  principle  of  the  gear-cutter 
series  may  be  illustrated  as  follows.  Suppose  we  wish 
to  construct  a  set  of  cutters  for  a  No.  i  pitch..  The 
extreme  variation  in  the  shape  of  the  cutters  must  obvi- 
ously be  between  the  cutter  for  the  gear  having  the 
greatest  diameter  (the  rack)  and  that  for  the  gear  having 
the  smallest  diameter  (say  the  pinion  having  eleven 
teeth).  Between  these  two  we  must  have  a  sufficient 
number  of  cutters  to  cut  No.  I  teeth  for  a  gear  of  any 
diameter  without  serious  error.  Similarly,  for  each  other 
necessary  pitch,  we  must  have  a  set  of  cutters  composed 
of  a  sufficient  number  to  make  our  errors  unimportant. 
Of  course  the  greater  number  of  cutters  we  have  in  each 
set,  the  more  accurate  will  be  our  work.  Thus,  if  we 
have  a  cutter  of  each  pitch  for  a  gear  of  eleven  teeth, 
another  for  a  gear  of  twelve  teeth,  another  for  thirteen 
teeth,  and  so  on,  our  gears  will  be  theoretically  accurate. 


TOOTHED    GEARING.  22 / 

But  gear-cutters  are  expensive  tools,  and  it  is  therefore 
important  to  reduce  the  number  to  the  minimum  which 
can  be  used  without  making  the  errors  so  great  as  to  do 
practical  harm.  Mr.  George  B.  Grant,  in  an  article  pub- 
lished some  months  ago  in  the  "American  Machinist," 
points  out  the  fact,  that,  since  the  extreme  variation  in 
the  shape  of  the  cutters  is  less  for  fine  than  for  coarse 
pitches,  the  number  of  cutters  necessary  for  the  same 
degree  of  accuracy  is  less  in  the  former  than  in  the 
latter.  He  gives  for  the  proper  number  of  cutters  in 
the  different  sets  the  following  table  :  — 

For  a  1 6  pitch  or  finer,    6  cutters 

For  an  8  to  1 6  pitch,                1 2  cutters 

For  a    4  to  8  pitch,                24  cutters 

For  a    2  to  4  pitch,               48  cutters. 

If  we  substitute  for  the  circumferential  pitch,  /  in 
formula  (10),  its  value  in  terms  of  the  diametral  pitch, 

L      v       3-i4i59\' 

\p=j*=-*r) 

we  shall  have 


or 


From  this,  by  transposing,  we  have, 


or  

"*£       (')• 


228  T007WED   GEARING. 

Rule.  —  To  find  the  diametral  pitch  for  a  gear  of 
any  material,  divide  the  greatest  safe  working-stress  in 
pounds  per  square  inch  for  the  material  used  by  the 
force  transmitted,  multiply  the  quotient  by  the  assumed 
ratio  of  the  face  width  to  the  circumferential  pitch,  ex- 
tract the  square  root  of  the  product  thus  obtained,  and 
multiply  the  result  by  0.637. 

By  substituting 

.        *•  =  3^4159 
P       Pd  pd 

in  formulas  (12,  at  b,  c\  we  obtain, 


,  =  3.14.59        yj? 
pd      p.t 

and 


From  these,  by  transposing,  we  have, 
_  3.14159. /i 


*-*= 

and 


^3-I4i594/i 

*  :  0.035  \  p 

or,  reducing  the  three  last  found  equations;,  we  have, 


TOOTHED   GEARING.  2 29 

For  violent  shock,       pd=  ST-12\  ~p    (a) 

i — 
For  moderate  shock,  pd—  62.83^  -p  (b) 

i — 
For  little  or  no  shock,  pd  =  89. 76V/  —  (c) 

Formulas  (12,  a,  b,  c)  were  determined  upon  the 
condition  that  the  face  width  equals  twice  the  circumfer- 
ential pitch :  hence  substituting/  =  —  in  the  expression 

Pd 
l—2p  gives, 

27T  6.283 

After  determining  the  diametral  pitch  pd  from  formula 

(2),  the  face  width  must  not  be  taken  less  than  . 

Pd 

Ride.  —  To  determine  the  diametral  pitch  for  a  cast- 
iron  gear,  when  /  =  -I—I,  extract  the  square  root  of 

Pd 

the  reciprocal  of  the  force  transmitted,  and  multiply 
the  result  by  57.12  for  violent  shock,  62.83  for  moder- 
ate shock,  or  89.76  for  little  or  no  shock. 

The  above  value  of  the  circumferential  in  terms  of  the 
diametral  pitch,  substituted  in  formulas  (14,  a,  b,  c),  gives 


^-'W? 


pd 


*   =  3-I4'59  =   Lx-i/fl 

Pd  Pd  '    V    V 

and 


Pd         Pd 


3.14159  =  0.82^. 


230  TOOTHED   GEARING. 

By  transposing, 


and 


or,  reducing,  we  have, 

For  violent  shock,        pd=  2.435X7  -^    (a) 

f     77 

For  moderate  shock,  /«?=  2.685X7^    (^) 


For  little  or  no  shock,  pd  =  3-83  iy  -73.     (<:) 


(3). 


As  before,  the  condition  /  ^  -^-     must  be  fulfilled. 

/^ 

Rule.  —  To  determine  the  diametral  pitch  for  a  cast- 
iron   gear   from   the   horse-power    and    circumferential 

velocity  in  feet  per  second,  when  /  =  ,  divide  the 

Pd 

velocity  by  the  horse-power,  extract  the  square  root  of 
the  quotient,  and  multiply  the  result  by  2.435  f°r  v^°- 
lent  shock,  2.685  f°r  moderate  shock,  or  3.831  for  little 
or  no  shock. 

In  a  similar  manner,  by  substituting  p  =.  —  in  formulas 

Pd 
(16,  a,  b,  c),  we  obtain, 


TOOTHED   GEARING. 

H 


and 


pd 


Transposing  and  reducing  these  three  equations,  as  with 
the  preceding,  we  have 


For  violent  shock,      pd  =  o.i6i\/-^     (a) 

-  _ 


For  moderate  shock,  pd  =  o.  1  7  yV   -^ 

.// 

For  little  or  no  shock,  /^  =  0.253^^ 


(4). 


?.  —  To  determine  the  diametral  pitch  for  a  cast- 
iron  gear  from  the  horse-power  and  number  of  revolu- 

tfans  per  minute,  when  /  =  -—  ^,  multiply  the  diameter 

Pd 

of  the  gear  by  the  number  of  revolutions,  divide  thc 
product  by  the  horse-power,  extract  the  square  root  of 
the  quotient  thus  obtained,  and  multiply  the  result  by 
o.  161  for  violent  shock,  0.177  for  moderate  shock,  or 
0.253  for  little  or  no  shock. 

Example  i.  —  Required  the  diametral  pitch  for  a  steel 
gear  which  will  transmit  a  force  of  30,000  pounds,  as- 

suming -=13;   the   greatest   safe  working-stress    per 
P 


232  TOOTHED    GEARING. 

square  inch  being  20,000  pounds.     From  formula  (i) 
we  have, 

/- 

=  0.637^2  =  0.637  x  1.41  -0.898. 


2OOOO 


Example  2.  —  Required  the  diametral  pitch  for  a  cast- 
iron  gear  to  transmit  a  force  of  900  pounds,  moderate 
shock.  From  formula  (2,  b)  we  have, 

/~7~      62.83 

pd  —  62.83V/     ~  = =  2-°9- 

^V  900         30 

Hence 

/=  6.283  =  6.283  =    ,/ 
pd      ~  2.09 

Example  3.  —  The  horse-power  to  be  transmitted  by 
a  cast-iron  gear  is  10,  moderate  shock,  and  the  circum- 
ferential velocity  5  feet  per  second.  Required  the 
diametral  pitch.  Formula  (3,  b)  gives 


•  '-90  • 


Arms  :  If,  in  formula  (23),  we  substitute  for/  its  value 
of  —  ,  we  will  have 

pd 


Extracting  the  fourth  root  of  •*  in  this  equation  gives 

x  i. 


or  _ 

I      ~~"         */T'       '  V      •ft.  \JJ* 


TOOTHED   GEARING.  233 

Rule.  —  To  determine  the  number  of  arms  in  a  gear, 
extract  the  square  root  of  the  number  of  teeth  and  the 
fourth  root  of  the  reciprocal  of  the  diametral  pitch  ; 
multiply  these  two  roots  together,  and  their  product  by 
0.746. 

From  the  expression  /  =  —  t-^  we  may  obtain,  by 

Pd 

squaring  both  sides,  p2  =  —  —  5-A     This,  substituted  in 

Pd 
formula  (29),  gives 

-  °'S  x  9.86965;? 


or 


Rule.  —  To  determine  the  quantity  bji?  (the  thick- 
ness of  the  arm  multiplied  by  the  square  of  the  width), 
divide  the  radius  of  the  gear  by  the  product  of  the 
square  of  the  diametral  pitch  into  the  number  of  arms, 
and  multiply  the  quotient  by  7.896. 

By  substituting/2  =  —  —  ,  in  formula  (30),  we  ob- 
tain, 


d'  =  1.105^/9.86965^-, 


which  reduces  to 


Rule.  —  To  determine  the  diameter  for  arms  having 
circular  cross-sections,  divide  the  radius  of  the  gear  by 
the  product  of  the  square  of  the  diametral  pitch  into  the 
number  of  arms,  extract  the  cube  root  of  the  quotient, 
and  multiply  the  result  by  2.37. 


234  TOOTHED   GEARING. 

In  a  similar  manner  we  may  obtain  from  formulas 
(31),  (32),  and  (33),  the  expressions, 

D 
b'a*  =  1.356  X  9.86965--^-, 

b,,H'*  +  ££,,3  7? 

-7T-  —  0.8x9.86965—, 

and 

BH'*  -  b,,h,t  R 

--       =  o.8x9.86965 


/—  , 
which  reduce  respectively  to  the  following  :  — 


and 

BH'*  -  b.,hj 


Rim,  Nave,  etc.  :  The  total  rim  thickness  before  the 
spaces  between  the  teeth  are  cut  out  is  equal  to  /  +  //- 
If  we  add  together  formula  (34)  and  the  expression  for 
the  total  height  of  the  teeth,  //  =  o.//,  we  shall  have, 


and,  by  substituting  for/  its  value  of 


=  0:12 


Pd 
0-7  X  3.14159 


pd  Pd 


*  See  Fig.  82.  t  See  Fig.  83. 


TOOTHED   GEARING.  235 

Or,  calling  the  total  thickness  of  the  rim  /,  and  reducing, 
we  obtain 

(i,). 


. 

Rule.  —  To  determine  the  total  thickness  of  the  rim 
(the  height  of  the  teeth  plus  the  true  rim  thickness), 
divide  3.46  by  the  diametral  pitch,  and  to  the  quotient 
add  o.i  2". 

The  expression  p2  —  —  —  ^-^,  substituted  in  formula 

Pd 
(35)i  Sives 

,3/9.8696 

=  °4V       /~  +    =  °4  X 


.86965^  R 

^~ 
or 

*  =  °.858^|  +  i        (12). 

Rule.  —  To  determine  the  thickness  of  the  nave, 
divide  the  radius  of  the  gear  by  the  square  of  the  di- 
ametral pitch,  extract  the  cube  root  of  the  quotient, 
multiply  the  root  by  0.858,  and  to  the  result  add  J  inch. 

For  the  length  of  the  nave  we  have  formula  (36), 
which  is, 


Formula  (45)  becomes,  on  substituting  for  p*  its  value 
in  terms  of  the  diametral  pitch, 


which  reduces  to 


236  TOOTHED   GEARING. 

Rule.  —  To  determine  the  diameter  of  a  gear  shaft  of 
any  material,  divide  the  radius  of  the  gear  by  the  prod- 
uct of  the  square  of  the  diametral  pitch  into  the  greatest 
safe  shearing-stress  in  pounds  per  square  inch  for  the 
material  of  the  shaft,  extract  the  cube  root  of  the  prod- 
uct thus  obtained,  and  multiply  the  result  by  27.184. 

Similarly  we  may  obtain  from  formulas  (46),  (47),  and 
(48)  the  equations, 


8/  R 

^=0-553^9-86965— 


</=  0.634^9-86965^ 
and  _ 

^=0.796^9.86965^. 
From  which,  by  reducing,  we  obtain  the  following  :  — 


For  steel,  ^=  i.i86y-2         (15) 

For  wrought-iron,  d  =  i-36y  —  (16) 

/-ft 

For  cast-iron,         ^=1.707^7- 


Pd 

Rule. — To  determine  the  diameter  of  a  gear  shaft, 
divide  the  radius  of  the  gear  by  the  square  of  the  diame- 
tral pitch,  extract  the  cube  root  of  the  quotient,  and 
multiply  the  result  by  1.186  for  steel,  1.36  for  wrought- 
iron,  or  1.707  for  cast-iron. 

The  formulas  for  the  mean  width  and  thickness  of 


TOOTHED   GEARING. 

the  fixing-key  are,  as  before  explained  for  formulas  (49) 
and  (50), 

S=o.i6  +  ™          (18) 
and 

S>=o.i6+*Q        (19). 

Example  4.  —  Required  to  design  a  24"  cut  gear-wheel 
(of  cast-iron)  which  will  safely  transmit  a  force  of  1,000 
pounds,  moderate  shock. 

From  formula  (2,  b)  we  have,  for  the  diametral  pitch, 

/    i         62.83 

pd  —  6  2.83V/  -   ~  ==  —  ~^~  =  2  verY  nearly. 
°V  1000      31.62 

The  face  width  is  consequently 


For  the  number  of  teeth  in  the  gear  we  have  the  ex- 
pression, 


N  =  pdD=  2  x  24  =  48. 
Therefore  formula  (5)  gives,  for  the  number  of  arms, 

n'  —  0.746^48  Vj  =  0.746  X  6.928  X  -  -  =  4. 

If  we  wish  to  have  rectangular  cross-sections  for  our 
arms,  and  take  the  thickness  equal  to  one-half  the  width, 
formula  (6)  gives 

hf      7-896  X  12 

O  !/J  ,2    =    ----    =    -  . 

2  4X4 

Hence 

h,  =  ^  1  1.  844  =  2.28" 
and 

2.28  „ 

" 


238  TOOTHED   GEARING. 

From  formula  (11)  we  have,  for  the  total  thickness  of 
the  rim, 

/'  =  0.12  +  — -  =  0.12  +  1.73  =  1.85". 

he  thickness  of  the  nave  is,  from  formula  (12), 

k  =  0.858^-  +  |  =  0.858  x  1.44  +  |  =  1.736" 
and  the  length,  from  formula  (13),  is 

/'  =  3.14  +  |-J  =  3.94". 

Formula  (16)  gives,  for  the  diameter  of  the  wrought- 
iron  shaft, 

d=  1.36?^-  =  1.36  x  1.44  =  1.96",  say  2". 

Formulas  (18)  and  (19)  give,  for  the  mean  width  and 
thickness  of  the  fixing-key, 

5=0.16+  f  =0.56" 
and 

5' =0.1 6+ ^  =  0.36". 

The  following  table  will  be  found  convenient  in 
constructing  cut  gears  of  cast-iron.  To  illustrate  its 
application,  suppose  we  have  to  construct  a  cut  gear 
which  will  transmit  a  force  of  4,000  pounds,  moderate 
shock.  We  find  in  the  table,  column  for  moderate 
shock,  P  =  3,948  pounds,  which  corresponds  to  a  No.  I 
diametral  pitch.  We  also  find  in  the  table  the  face 
width  of  6.28",  and  the  total  rim  thickness  of  3.58". 


TOO  7^HED   GEA RING. 


239 


"V 

V 

P 

P 

P 

77 

H 

77 

/' 

/ 

p* 

Moder- 

Little 

Little 

Violent 

Violent 

Moderate 

In 

shock. 

ate 
shock. 

or  no 
shock. 

shock. 

shock. 

or  no 
shock. 

inches. 

In  inches. 

i 

52203 

63l66 

128910 

0.0106 

0.0087 

0.0043 

13.96 

25-I3 

! 

23201 

28074 

57293 

0.0237 

0.0195 

0.0095 

9-36 

16.76 

i 

I305I 

I579I 

32227 

0.0423 

0.0350 

0.0170 

7.04 

12.57 

1 

8352 

IOI06      20625 

0.066 

0.054 

0.027 

5.66 

10.05 

I 

5800 

7018 

H323 

0.095 

0.078 

0.038 

4-73 

8.38 

i 

3263 

3948 

8057 

0.169 

0.139 

0.068 

3-58 

6.28 

'i 

2088 

2527 

5156 

0.265 

0.217 

0.107 

2.89 

5-03 

ii 

1450 

1754 

3581 

0.380 

0.312 

0.153 

243 

4.19 

'i 

1066 

1290 

2631 

0.519 

0.425 

0.208 

2.10 

3-59 

2 

816 

987 

2014 

0.676 

o-555 

O.272 

1.85 

3-14 

a* 

644 

779 

1591 

0.858 

0.702 

0-344 

1.66 

2.79 

4 

522 

632 

I289 

1.056 

0.867 

0.425 

1.50 

2.51 

2} 

43i 

522 

1065 

1.283 

1.049 

0.514 

1-37 

2.28 

3 

362 

439 

895 

I.52I 

1.248 

0.612 

1.27 

2.09 

4 

204 

247 

504 

2.704 

2.219 

1.  088 

0.99    1.57 

5 

131 

158 

322 

4.225 

3467 

1.700 

0.81 

1.26 

6 

9i 

no        224 

6.084 

4-993 

2.448 

0.70 

1.05 

7 

67 

81 

164 

9.281 

6.796 

3-332 

0.61 

0.90 

8 

5i 

62 

126 

10.816 

8.877 

4-352 

0.55 

0.79 

9 

40 

49 

99 

13.689 

11.235 

5.508 

0.50 

0.70 

10 

33 

40 

8r 

16.863 

13-870 

6.812 

0.47 

0.63 

12 

23 

27 

56 

24-336 

19-973 

9.792 

0.41 

0.52 

INDEX. 


[The  numbers  refer  to  the  pages.] 


A. 


Actual  pitch,  52. 
Angle  of  repose,  57. 
Arc  of  approach,  89. 
of  contact,  44,  89. 
of  recess,  89. 
Arms,  circular  sections,  no. 

curved,  122. 

elliptical  sections,  HI. 

flanged  sections,  112. 

methods  for  drawing,  122. 

number  of,  115. 

rectangular  sections,  108. 

straight,  122. 

strength  of,  107. 

B. 

Bastard  gears,  54. 
Bevel  gears,  49. 

angle  of  shafts  of,  49. 

design  of,  154. 

drawings  of,  160. 

internal,  53. 

method  for  drawing,  52. 

mutilated,  213. 

scroll,  214. 
Bevel  rack,  54. 
Breadth  of  teeth,  89,  97. 
Breaking-weight,  96. 


C. 


Cam-pinion,  206. 

Circle,  generating,  15,  20,  33. 

of  centres,  32. 

of  the  gorge,  68. 

pitch,  15,  17,49. 

primitive,  17,  19. 

rolling,  ii,  28,  31. 

root,  36. 

top,  36. 

Circumference,  72. 
Circumferential  pitch,  73. 
Conditions  for  minimum  friction,  icx 
for  uniform  velocity,  1 5. 
Cone,  pitch,  49. 

supplementary,  50. 
Constant  TT,  72. 
Crown  gears,  210.    . 
Cycloid,  37. 
Cycloidal  teeth,  22. 
Cylindrical  gears,  49,  54. 

D. 

Decimals,  table  of,  106. 
Design  of  bevel  gears,  154. 

of  gear  train,  186. 

of  internal  lantern,  183. 

of  internal  spur  gear,  169. 

of  1-antern  gear,  180. 
241 


242 


INDEX. 


Design  of  rack  and  pinion,  175. 
of  screw  gears,  164. 
of  spur  gear,  151. 
of  worm  and  wheel,  160. 
Diameter,  72. 

Diametral  formulas,  arms,  232. 
cutters,  227. 
nave,  234. 
rim,  234. 
shafts,  235. 
Diametral  pitch,  74. 
Dimensions  for  bevel  gears,  158. 
for  gear  train,  194. 
for  internal  lantern,  185. 
for  internal   spur  gear, 

174. 

for  lantern  gear,  182. 
for  rack  and  pinion,  178. 
for  screw  gears,  168. 
for  spur  gear,  153. 
for  worm    and   wheel, 

163. 

Disk  wheel,  53. 
Drawings  of  bevel  gears,  160. 
of  gear  train,  196. 
of  internal  lantern,  187. 
of  internal  spur  gear,  176. 
of  lantern  gear,  184. 
of  rack  and  pinion,  179. 
of  screw  gears,  170. 
of  spur  gear,  1 55. 
of  worm  and  wheel,  164. 

E. 

Elliptical  gears,  201. 
Epicycloid,  u,  17. 
Epicycloidal  faces,  13,  16. 
Examples,  arms,  109-120. 

bevel  gears,  154. 

diameter,  80. 

face  width,  97-100. 

gear  train,  186. 


Examples,  hyperbolic  gears,  68. 

internal  "lantern,  183. 

internal  spur  gear,  169. 

keys,  136. 

lantern  gear,  180. 

nave,  126. 

number  of  teeth,  80. 

pitch,  74,  96-100. 

pitch,  diametral,  74. 

power,  84. 

rack  and  pinion,  175. 

revolutions,  81. 

rim,  125. 

screw  gears,  164. 

shafts,  128-135. 

spur  gear,  151. 

velocity,  84. 

weight  of  gears,  137. 

worm  and  wheel,  160. 
Experiments  v/ith  involute  teeth,  24. 

F. 

Face,  epicycloidal,  13,  16, 
involute,  22. 
width,  89-93. 

Flank,  hypocycloidal,  13,  16. 
radial,  19,  48. 
straight,  19,  48. 
Formulas  for  arms,  circular,  no,  117, 

1 20. 
for   arms,    elliptical,  in, 

118,  121. 

for  arms,  flanged,  112,  114, 

119,  121. 

for  arms,  rectangular,  108, 

116,  120. 

for  chord  of  the  pitch,  75. 
for  circumference,  72. 
for  diameter,  72. 
for  diametral  pitch,  73. 
for  fixing-keys,  136. 
for  nave  kugth,  126. 


INDEX. 


243 


Formulas  for  nave  thickness,  125. 

for  number  of  arms,  115. 

for  number  of  revolutions, 
80. 

for  number  of  teeth,  73. 

for  pitch,  from  force  trans- 
mitted, 91-93- 

for  pitch,  from  horse-pow- 
er, 94-96. 

for   pitch,    from    revolu- 
tions, 95. 

for  power,  83. 

for  radius,  72. 

for  rim,  125. 

for  shafts,  127-131. 

for  velocity,  83. 

for  weight  of  gears,  136. 
Fractions,  table  of,  106. 
Friction,  minimum,  10. 
Fundamental  principle,  2. 

G. 

Gears,  bastard,  54. 
bevel,  49. 
cast,  224. 
crown,  210. 
cut,  226. 
cylindrical,  54. 
elliptical,  201. 
high-speed,  107. 
hyperbolic,  65. 
internal,  40,  169. 
lantern,  43. 
mangle,  216. 
mixed,  47. 
mutilated,  208,  214. 
pin,  212. 
rectangular,  198. 
screw,  54. 
scroll,  202. 
sector,  204. 
spur,  49. 


Gears,  square,  198. 
stepped,  206. 
triangular,  200. 
Gear  at  two  points,  46. 
Generating  circle,  15,  17,  33. 
Generating  of  epicycloid,  n. 
of  hypocycloid,  i: 
of  involute,  21. 

H. 

Height  of  teeth,  90. 
working,  35. 
High-speed  gears,  107. 
Horse-power,  94. 
Hyperbolic  gears,  65. 
Hypocycloid,  12,  18. 
Hypocycloidal  flanks,  13,  16. 

I. 

Infinite  radius,  28,  37. 
Intermittent  motion,  205. 
Internal  bevels,  53. 

lantern  gears,  44. 

spur  gears,  169. 

worm  wheel,  62. 
Introduction,  i. 
Involute,  21. 

faces,  22. 
profiles,  21. 
Irregular  motion,  208. 

K. 

Keys,  formulas  for,  136. 
rules  for,  136. 

L. 

Lantern  gears,  43. 

internal,  44. 
Line  of  contact,  87. 


M. 


Mangle  wb«el, 


244 


INDEX. 


Method  for  drawing  bevels,  52. 

for   drawing   curved   arms, 

122. 

for  drawing  cycloidal   pro- 
files, 28,  31. 

for   drawing    involute   pro- 
files, 35. 
for  stepping  off  the  pitch, 

76. 

Minimum  friction,  10. 
Mixed  gears,  47. 
Motion,  intermittent,  205. 
irregular,  208. 
quick  return,  207. 
reciprocating,  208. 
rectilinear,  197. 
rotary,  3,  198. 
\     uniform,  10. 
variable,  212. 

N. 

Nave,  125. 
Notation,  139. 
Number  of  arms,  1 1 5. 

of  revolutions,  80. 

of  teeth,  73. 

P. 

Pcricycloid,  45. 
Pin  wheel,  212. 
Pi-rule,  77. 
Pitch,  actual,  52. 

circle,  15,  17,  49. 

circumferential,  73. 

cone,  49. 

diametral,  74. 

frusta,  49. 

point,  15. 

virtual,  52. 
Plane  wheel,  53. 
Power  ratio,  82. 
Primitive  circle,  17,  19. 


Primitive  gear  wheel,  4,  7. 
Profiles,  cycloidal,  37. 

epicycloidal,  13. 

hypocycloidal,  13. 

involute,  21. 

Q- 

Quick  return  motion,  207. 

R. 

Rack,  37. 
Radius,  72. 

infinite,  28,  37. 
Ratio,  power,  82. 

revolution,  79. 

velocity,  78. 
Recapitulation,  139. 
Reciprocating  motion,  208. 
Rectilinear  motion,  197. 
Rolling  circle,  n. 
Root  circle,  36. 
Rotary  motion,  3,  198. 
Rules  for  arms,  circular,  in,  117. 

for  arms,  elliptical,  in,  118. 

for  arms,  rectangular,  109. 

for  circumference,  72. 

for  diameter,  72. 

for  fixing-keys,  136. 

for  nave  length,  126. 

for  nave  thickness,  125. 

for  number  of  arms,  115. 

for  number  of  revolutions,  So. 

for  number  of  teeth,  73. 

for  pitch,  from  force  transmit- 
ted, 92,  93. 

for  pitch,  from  horse-power, 

94,  95- 
for  pitch,  from  revolutions, 

95,96. 

for  power,  84. 
for  radius,  72. 
for  rim,  125. 


INDEX. 


245 


Rules  for  shafts,  127,  130. 

for  weight  of  gears,  136. 

S. 

Safe  shearing-stress,  127. 

working-stress,  90. 
Screw  gears,  54. 
rack,  58. 

Scroll  gears,  202. 
Sector  gears,  204. 
Shafts,  cast-iron,  132. 

formulas  for,  127,  131. 
rules  for,  127,  131. 
steel,  132. 
tables  for,  133. 
wrought-iron,  132. 
Special  forms,  45. 
Spur  gears,  49. 
Square  gears,  198. 
Stepped  gears,  206. 
Straight  flanks,  19,  48. 
Strength  of  arms,  107. 
of  keys,  136. 
of  nave,  125. 
of  rim,  125. 
of  shafts,  127. 
of  teeth,  89. 

Supplementary  angle,  66. 
cones,  50. 

T. 

Tables  for  arm  widths,  109. 

for  decimals   and    fractions, 

106. 
for  diametral  pitches,  224. 


Tables  for  number  of  arms,  1 1 5. 

for  number  of  gear  cutters, 
227. 

for  pitch,  101. 

for  shaft  diameters,  133. 

for  weight  of  gears,  138. 
Teeth,  cast,  224. 

cut,  226. 

cycloidal,  22. 

involute,  22. 

of  bevels,  52. 

of  hyperbolic  gears,  71. 

of  screw  gears,  60. 
Top  circle,  36. 
Train  of  gears,  81,  186. 
Triangular  gears,  200. 

U. 

Uniform  motion,  10. 
velocity,  15. 

V. 

Variable  motion,  212. 
Velocity  ratio,  78. 
Virtual  pitch,  52. 

W. 

Wear  on  teeth,  8,  63. 
Weight  of  gears,  136. 
Working  height,  35. 
stress,  90. 

Worm,  internal,  62. 
and  rack,  62. 
and  wheel,  61. 


SHORT-TITLE   CATALOGUE 

OP  THE 

PUBLICATIONS 

OF 

JOHN   WILEY   &    SONS, 

NEW    YORK, 
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All  books  are  bound  in  cloth  unless  otherwise  stated. 


AGRICULTURE. 

Arinsby 's  Manual  of  Cattle  Feeding 12mo,  $1  75 

Downing's  Fruit  and  Fruit  Trees 8vo,  5  00 

Grotenfelt's  The  Principles  of  Modern  Dairy  Practice.     (Woll.) 

12mo,  2  00 

Kemp's  Landscape  Gardening 12mo,  2  50 

Maynard' s  Landscape  Gardening 12mo,  1  50 

Steel's  Treatise  on  the  Diseases  of  the  Dog 8vo,  3  50 

Stockbridge's  Rocks  and  Soils 8vo,  2  50 

Woll's  Handbook  for  Farmers  and  Dairymen 12mo,  1  50 

ARCHITECTURE. 

Berg's  Buildings  and  Structures  of  American  Railroads 4to,  5  00 

Birkmire's  American  Theatres— Planning  and  Construction,  8vo,  3  00 

"        Architectural  Iron  and  Steel 8vo,  3  50 

"        Compound  Riveted  Girders 8vo,  2  00 

"        Skeleton  Construction  in  Buildings 8vo,  3  00 

"        Planning  and  Construction  of  High  Office  Buildings. 

8vo,  3  50 

Briggs'  Modern  Am.  School  Building 8vo,  4  00 

Carpenter's  Heating  and  Ventilating  of  Buildings 8vo,  3  00 

1 


Freitag's  Architectural  Engineering 8vo,  f  2  50 

The  Fireproofiug  of  Steel  Buildings 8vo,  2  50 

Gerhard's  Sanitary  House  Inspection 16mo,  1  00 

Theatre  Fires  and  Panics 12mo,  1  50 

Hatfield's  American  House  Carpenter 8vo,  5  00 

Holly's  Carpenter  and  Joiner 18mo,  75 

Kidder's  Architect  and  Builder's  Pocket-book. .  .16mo,  morocco,  4  00 

Merrill's  Stones  for  Building  and  Decoration 8vo,  5  00 

Monckton's  Stair  Building — Wood,  Iron,  and  Stone 4to,  4  00 

Wait's  Engineering  and  Architectural  Jurisprudence 8vo,  6  00 

Sheep,  6  50 

"       Law  of  Operations  Preliminary  to  Construction  in  En- 
gineering and  Architecture. . .   8vo,  5  00 

Sheep,  5  50 

Worcester's  Small  Hospitals — Establishment  and  Maintenance, 
including  Atkinson's  Suggestions  for  Hospital  Archi- 
tecture  12mo,  1  25 

*  World's  Columbian  Exposition  of  1893 Large  4to,  1  00 

ARMY,  NAVY,  Etc. 

Bernadou's  Smokeless  Powder 12mo,  2  50 

*Bruffs  Ordnance  and  Gunnery 8vo,  6  00 

Chase's  Screw  Propellers 8yo,  3  00 

Cronkhite's  Gunnery  for  Non-com.  Officers 32mo,  morocco,  2  00 

*  Davis's  Treatise  on  Military  Law 8vo,  7  00 

Sheep,  7  50 

*  "       Elements  of  Law 8vo,  2  50 

De  Brack's  Cavalry  Outpost  Duties.     (Carr.). .  .  .32mo,  morocco,  2  00 

Dietz's  Soldier's  First  Aid 16mo,  morocco,  1  25 

*  Dredge's  Modern  French  Artillery..  ..Large  4to,  half  morocco,  15  00 

*  "          Record   of   the   Transportation    Exhibits    Building, 

World's  Columbian  Exposition  of  1893.. 4to,  half  morocco,  5  00 

Durand's  Resistance  and  Propulsion  of  Ships 8vo,  5  00 

*  Fiebeger's    Field   Fortification,    including  Military  Bridges, 

Demolitions,  Encampments  and  Communications. 

Large  12mo,  2  00 

*Dyer's  Light  Artillery 12mo,  3  00 

*Hoff  s  Naval  Tactics. 8vo,  1  50 

*Ingalls's  Ballistic  Tables 8vo,  1  50 

2, 


Ingalls's  Handbook  of  Problems  in  Direct  Fire 8vo,  $4  00 

Mahau's  Permanent  Fortifications.  (Mercur.).8vo,  half  morocco,  7  50 

Manual  for  Courts-Martial 16mo,  morocco,  1  50 

*  Mercurs  Attack  of  Fortified  Places 12mo,  2  00 

*  "        Elements  of  the  Art  of  War 8vo,  4  00 

Metcalfe's  Ordnance  and  Gunnery 12mo,  with  Atlas,  5  00 

Murray's  Infantry  Drill  Regulations  adapted  to  the  Springfield 

Rifle,  Caliber  .45 32mo,  paper,  10 

*  Phelps's  Practical  Marine  Surveying 8vo,  2  50 

Powell's  Army  Officer's  Examiner 12mo,  4  00 

Sharpe's  Subsisting  Armies 32mo,  morocco,  1  50 

Wheeler's  Siege  Operations 8vo,  2  00 

Wiuthrop's  Abridgment  of  Military  Law 12mo,  2  50 

Woodhull's  Notes  on  Military  Hygiene 16mo,  1  50 

Young's  Simple  Elements  of  Navigation. 16mo,  morocco,  2  00 

first  edition 100 

ASSAYING. 

Fletcher's  Quant.  Assaying  with  the  Blowpipe..  16mo,  morocco,  1  50 

Furman's  Practical  Assaying , 8vo,  3  00 

Kuuhardt's  Ore  Dressing 8vo,  1  50 

Miller's  Manual  of  Assaying 12mo,  1  00 

O'Driscoll's  Treatment  of  Gold  Ores 8vo,  2  00 

Ricketts  and  Miller's  Notes  on  Assaying 8vo,  3  00 

Thurston's  Alloys,  Brasses,  and  Bronzes 8vo,  2  50 

Wilson's  Cyanide  Processes 12mo,  1  50 

"       The  Chlorination  Process 12mo,  150 

ASTRONOMY. 

Craig's  Azimuth 4to,  3  50 

Doolittle's  Practical  Astronomy 8vo,  4  00 

Gore's  Elements  of  Geodesy 8vo,  2  50 

Hay  ford's  Text-book  of  Geodetic  Astronomy 8vo.  3  00 

*  Michie  and  Harlow's  Practical  Astronomy 8vo,  3  00 

*  White's  Theoretical  and  Descriptive  Astronomy 12mo,  2  00 

BOTANY. 

Baldwin's  Orchids  of  New  England Small  8vo,  1  50 

Thome's  Structural  Botany 16mo,  2  25 

Westermaier's  General  Botany.     (Schneider.) 8vo,  2  00 

3 


BRIDGES,  ROOFS,   Etc. 

(See  also  ENGINEERING,  p.  7.) 

Boiler's  Highway  Bridges 8vo,  $2  00 

*  "      The  Thames  River  Bridge 4to,  paper,  5  00 

Burr's  Stresses  iii  Bridges. ...   8vo,  3  50 

Crehore's  Mechanics  of  the  Girder 8vo,  5  00 

Du  Bois's  Stresses  in  Framed  Structures Small  4to,  10  00 

Foster's  Wooden  Trestle  Bridges 4to,  5  00 

Greene's  Arches  in  Wood,  etc 8vo,  2  50 

Bridge  Trusses 8vo,  250 

Roof  Trusses 8vo,  125 

Howe's  Treatise  on  Arches 8vo,  4  00 

Johnson's  Modern  Framed  Structures Small  4to,  10  00 

Merriman  &  Jacoby's  Text-book  of  Roofs  and  Bridges. 

Part  I.,  Stresses 8vo,  250 

Merriman  &  Jacoby's  Text-book  of  Roofs  and  Bridges. 

Part  II.,  Graphic  Statics. 8vo,  2  50 

Merrimau  &  Jacoby's  Text-book  of  Roofs  and  Bridges. 

Part  III.,  Bridge  Design 8vo,  2  50 

Merriman  &  Jacoby's  Text-book  of  Roofs  and  Bridges. 

Part  IV.,  Continuous,  Draw,  Cantilever,  Suspension,  and 

Arched  Bridges 8vo,  2  50 

*  Morison's  The  Memphis  Bridge Oblong  4to,  10  00 

Waddell's  De  Poutibus  (a  Pocket-book  for  Bridge  Engineers). 

16mo,  morocco,  3  00 

"       Specifications  for  Steel  Bridges 12mo,  125 

Wood's  Construction  of  Bridges  and  Roofs 8vo,  2  00 

Wright's  Designing  of  Draw  Spans.     Parts  I.  and  II..8vo,  each  2  50 

"               "          "      "           "          Complete 8vo,  350 

CHEMISTRY— BIOLOGY-PHARMACY— SANITARY  SCIENCE. 

Adriauce's  Laboratory  Calculations 12mo,  1  25 

Allen's  Tables  for  Iron  Analysis Svo,  3  00 

Austen's  Notes  for  Chemical  Students 12mo,  1  50 

Bolton's  Student's  Guide  in  Quantitative  Analysis 8vo,  1  50 

Classen's  Analysis  by  Electrolysis.   (Herrick  and  Boltwood.).8vo,  3  00 

4 


Conn's  Indicators  and  Test-papers » I2mo  $2  00 

Crafts's  Qualitative  Analysis.     (Schaeffer.) 12mo,  1  50 

Davenport's  Statistical  Methods  with  Special  Reference  to  Bio- 
logical Variations 12mo,  morocco,  1  25 

Drechsel's  Chemical  Reactions.    (Merrill.) 12mo,  1  25 

Erdmann's  Introduction  to  Chemical  Preparations.     (Duulap.) 

12mo,  1  25 

Fresenius's  Quantitative  Chemical  Analysis.    (Allen.) 8vo,  6  00 

Qualitative          "               "            (Johnson.) 8vo,  300 

(Wells.)         Trans. 

16th  German  Edition 8vo,  5  00 

Fuertes's  Water  and  Public  Health , .  12mo,  1  50 

Water  Filtration  Works 12mo,  2  50 

Gill's  Gas  and  Fuel  Analysis 12mo,  1  25 

Goodrich's  Economic  Disposal  of  Towns'  Refuse Demy  8vo,  3  50 

Hammarsten's  Physiological  Chemistry.    (Maudel.) 8vo,  4  00 

Helm's  Principles  of  Mathematical  Chemistry.    (Morgan).  12mo,  1  50 

Hopkins'  Oil- Chemist's  Hand-book 8vo,  3  00 

Ladd's  Quantitative  Chemical  Analysis 12mo,  1  00 

Landauer's  Spectrum  Analysis.     (Tingle. ) 8vo,  3  00 

Lob's  Electrolysis  and  Electrosyn thesis  of  Organic  Compounds. 

(Loreuz.) 12mo,  1  00 

Mandel's  Bio-chemical  Laboratory 12mo,  1  50 

Mason's  Water-supply 8vo,  5  00 

"      Examination  of  Water 12mo,  1  25 

Meyer's  Radicles  in  Carbon  Compounds.  (Tingle.) 12mo,  1  00 

Mixter's  Elementary  Text-book  of  Chemistry 12mo,  1  50 

Morgan's  The  Theory  of  Solutions  and  its  Results 12mo,  1  00 

Elements  of  Physical  Chemistry 12mo,  2  00 

Nichols's  Water-supply  (Chemical  and  Sanitary) 8vo,  2  50 

O'Brine's  Laboratory  Guide  to  Chemical  Analysis 8vo,  2  00 

Dinner's  Organic  Chemistry.     (Austen.) 12mo,  1  50 

Poole's  Calorific  Power  of  Fuels 8vo,  3  00 

Richards's  Cost  of  Living  as  Modified  by  Sanitary  Science..  12mo.  1  00 

"        and  Woodman's  Air,  Water,  and  Food 8vo,  2  00 

Ricketts  and   Russell's  Notes  on   Inorganic  Chemistry  (Non- 
metallic)  Oblong  8vo,  morocco,  75 

Rideal's  Sewage  and  the  Bacterial  Purification  of  Sewage... 8vo,  3  50 

Ruddimau's  Incompatibilities  in  Prescriptions 8vo,  2  00 

Schimpf's  Volumetric  Analysis 12mo,  2  50 

Spencer's  Sugar  Manufacturer's  Handbook 16mo,  morocco,  2  00 

"          Handbook    for    Chemists    of   Beet    Sugar    Houses. 

16mo,  morocco,  3  00 

Stockbridge's  Rocks  and  Soils 8vo,  2  50 

*  Tillman's  Descriptive  General  Chemistry 8vo,  3  00 

5 


Van  Deventer's  Physical  Chemistry  for  Beginners.    (Bolt  wood.) 

12mo,  $1  50 

Wells's  Inorgauic  Qualitative  Analysis 12ino,  1  50 

"      Laboratory  Guide  in   Qualitative  Chemical  Analysis. 

8vo,  1  50 

Whipple's  Microscopy  of  Drinkiug-water 8vo,  3  50 

Wiechmann's  Chemical  Lecture  Notes 12mo,  3  00 

Sugar  .Analysis Small  8vo,  2  50 

Wulling's  Inorganic  Phar.  and  Med.  Chemistry ISino,  2  00 

DRAWING. 

*  Bartlett's  Mechanical  Drawing 8vo,  3  00 

Hill's  Shades  and  Shadows  and  Perspective 8vo,  2  00 

MacCord's  Descriptive  Geometry 8vo,  3  00 

"          Kinematics 8vo,  500 

"           Mechanical  Drawing 8vo,  4  00 

Mahan's  Industrial  Drawing.    (Thompson.) 2  vols.,  8vo,  3  50 

Reed's  Topographical  Drawing.     (H.  A.) 4to,  5  00 

Reid's  A  Course  in  Mechanical  Drawing 8vo.  2  00 

"      Mechanical  Drawing  and  Elementary  Machine   Design. 

8vo,  3  00 

Smith's  Topographical  Drawing.     (Macmillan.) 8vo,  2  50 

Warren's  Descriptive  Geometry 2  vols.,  8vo,  3  50 

"        Drafting  Instruments 12mo,  1  25 

Free-hand  Drawing 12mo,  1  00 

"        Linear  Perspective 12mo,  1  00 

"        Machine  Construction 2  vols.,  8vo,  7  50 

Plane  Problems 12mo,  125 

"        Primary  Geometry 12mo,  75 

"        Problems  and  Theorems 8vo,  2  50 

"        Projection  Drawing 12mo,  150 

Shades  and  Shadows 8vo,  300 

"        Stereotomy — Stone-cutting ...8vo,  250 

Whelpley's  Letter  Engraving ...  12mo,  2  00 

Wilson's  Free-hand  Perspective 8vo,  2  50 

ELECTRICITY,  MAGNETISM,  AND  PHYSICS. 

Anthony  and  Brackett's  Text-book  of  Physics.     (Magie.)   Small 

8vo,  3  00 

Anthony's  Theory  of  Electrical  Measurements 12mo,  1  00 

Barker's  Deep-sea  Soundings 8vo,  2  00 

Benjamin's  Voltaic  Cell 8vo,  3  00 

History  of  Electricity 8vo,  300 

6 


Classen's  Analysis  by  Electrolysis.    (Jleriick  aud  Boltwuod  )  8vo,  $3  00 
Crehore  aud  Squier's  Experiments  with  a  New  Polarizing  Photo- 
Chronograph 8vo,  3  00 

Dawson's  Electric  Railways  and  Tramways.     Small,  4to,  half 

morocco,  12  50 
*"  Engineering  "  and  Electric  Traction  Pocket-book.      16mo, 

morocco,  5  00 

*  Dredge's  Electric  Illuminations.  .  .  .2  vols. ,  4to,  half  morocco,  25  00 

Vol.  II 4to,  7  50 

Gilbert's  De  magnete.     (Mottelay .) 8vo,  2  50 

Holman's  Precision  of  Measurements 8vo,  2  00 

"         Telescope-mirror-scale  Method Large  8vo,  75 

Le  Chatelier's  High  Temperatures.     (Burgess) 12mo,  3  00 

Lob's  Electrolysis  aud  Electrosyuthesis  of  Organic  Compounds. 

(Lorenz. ) 12mo,  1  00 

Lyous's  Electromagnetic  Phenomena  and  the  Deviations  of  the 

Compass 8vo,  6  00 

*Michie's  Wave  Motion  Relating  to  Sound  aud  Light 8vo,  4  00 

Morgan's  The  Theory  of  Solutions  and  its  Results 12mo,  1  00 

Niaudet's  Electric  Batteries      (Fishback.) 12mo,  250 

*Parshall  &  Hobart  Electric  Generators.     Small  4to,  half  mor.,  10  00 

Pratt  and  Alden's  Street-railway  Road-beds 8vo,  2  00 

Reagan's  Steam  and  Electric  Locomotives 12mo,  2  00 

Thurston's  Stationary  Steam  Engines  for  Electric  Lighting  Pur- 
poses    8vo,  2  50 

*Tillman's  Heat 8vo,  1  50 

Tory  &  Pitcher's  Laboratory  Physics Small  8vo,  2  00 


ENGINEERING. 

CIVIL — MECHANICAL— SANITARY,  ETC. 
See  also    BHIDGES,    p.   4 ;   HYDRAULICS,   p.   9 ;   MATERIALS  OP  EN- 

GINEERING,  p.    1 1  ;    MECHANICS  AND   MACHINERY,  p.  12  J  STEAM 

ENGINES  AND  BOILERS,  p.  14.) 

Baker's  Masonry  Construction .<...*..  8vo,  6  00 

"        Surveying  Instruments 12mo,  3  00 

Black's  U.  S.  Public  Works Oblong  4to,  5  00 

Brooks's  Street-railway  Location 16mo,  morocco,  1  50 

Butts's  Civil  Engineers'  Field  Book 16mo,  morocco,  2  50 

Byrne's  Highway  Construction 8vo,  5  00 

"       Inspection  of  Materials  and  Workmanship 16mo,  3  00 

Carpenter's  Experimental  Eugineering 8vo,  6  00 

Church's  Mechanics  of  Engineering — Solids  and  Fluids  —  8vo,  6  00 

7 


Church's  Notes  and  Examples  iu  Mechanics 8vo,  $2  00 

Oandall's  Earthwork  Tables 8vo,  1  50 

The  Transition  Curve 16mo,  morocco,  1  50 

Davis's  Elevation  and  Stadia  Tables Small  8vo,  1  00 

Dredge's     Penn.    Railroad     Construction,    etc.       Large     4to, 

half  morocco,  $10;  paper,  5  00 

*  Drinker's  Tunnelling 4to,  half  morocco,  25  00 

Eissler's  Explosives — Nitroglycerine  and  Dynamite 8vo,  4  00 

Frizell's  Water  Power 8vo,  5  00 

Folwell's  Sewerage 8vo,  3  00 

"        Water-supply  Engineering 8vo,  4  00 

Fowler's  Coffer-dam  Process  for  Piers . .  8vo.  2  50 

Fuertes's  Water  Filtration  Works 12mo,  2  50 

Gerhard's  Sanitary  House  Inspection 12mo,  1  00 

Godwin's  Railroad  Engineer's  Field-book 16mo,  morocco,  2  50 

Goodrich's  Economic  Disposal  of  Towns'  Refuse Demy  8vo,  3  50 

Gore's  Elements  of  Geodesy Svo,  2  50 

Hazlehurst's  Towers  and  Tanks  for  Cities  and  Towns Svo,  2  50 

Howard's  Transition  Curve  Field-book 16mo,  morocco,  1  50 

Howe's  Retaining  Walls  (New  Edition.) 12mo,  1  25 

Hudson's  Excavation  Tables.     Vol.  II r 8vo,  1  00 

Button's  Mechanical  Engineering  of  Power  Plants Svo,  5  00 

"        Heat  and  Heat  Engines Svo,  500 

Johnson's  Materials  of  Construction Svo,  6  00 

"         Theory  and  Practice  of  Surveying Small  Svo,  4  00 

Kent's  Mechanical  Engineer's  Pocket-book 16mo,  morocco,  5  00 

Kiersted's  Sewage  Disposal 12mo,  1  25 

Mahan's  Civil  Engineering.      (Wood.) Svo,  5  00 

Merriman  and  Brook's  Handbook  for  Surveyors 16mo,  nior.,  2  00 

Merriman's  Precise  Surveying  and  Geodesy Svo,  2  50 

"          Sanitary  Engineering Svo,  2  00 

Nagle's  Manual  for  Railroad  Engineers .16mo,  morocco,  3  00 

Ogdeu's  Sewer  Design 12mo,  2  00 

Pattou's  Civil  Engineering ,8vo,  half  morocco,  7  50 

Foundations Svo,  500 

Philbrick's  Field  Manual  for  Engineers 16mo,  morocco,  3  00 

Pratt  and  Aldeu's  Street-railway  Road-beds 8vo,  2  00 

Rockwell's  Roads  and  Pavements  in  France 12mo,  1  25 

Schuyler's  Reservoirs  for  Irrigation Large  Svo,  5  00 

Searles's  Field  Engineering = 16mo,  morocco,  3  00 

"       Railroad  Spiral 16mo,  morocco,  1  50 

Siebert  and  Biggin's  Modern  Stone  Cutting  and  Masonry. .  .Svo,  1  50 

Smart's  Engineering  Laboratory  Practice 12mo,  2  50 

Smith's  Wire  Manufacture  and  Uses Small  4to,  3  00 

Spalding's  Roads  and  Pavements 12mo,  2  00 

S 


Spalding's  Hydraulic  Ceineut I2mo,  $'2  00 

Taylor's  Prismoidal  Formulas  aud  Earthwork 8vo,  1  50 

Tim rst oil's  Materials  of  Construction  „ 8vo,  5  00 

Tillson's  Street  Pavements  and  Paving  Materials 8vo,  4  00 

*  Trautwiue's  Civil  Engineer's  Pocket-book 16mo,  morocco,  5  00 

*  •'          Cross-section Sheet,  25 

*  ' '           Excavations  and  Embankments 8vo,  2  00 

*  "            Laying  Out  Curves 12uio,  morocco,  2  50 

Ttirneaure  and  Kussell  s  Public  Water-supplies 8vo,  5  00 

Waddell's  De  Pontibus  (A  Pocket-book  for  Bridge  Engineers). 

16mo,  morocco,  3  00 

Wait's  Engineering  and  Architectural  Jurisprudence 8vo,  6  00 

Sheep,  6  50 

"      Law  of  Field  Operati      in  Engineering,  etc 8vo,  5  00 

Sheep,  5  50 

Warren's  Stereotomy— Stone-cutting 8vo,  2  50 

Webb's  Engineering  Instruments.  New  Edition.  16mo,  morocco,  1  25 

"      Railroad  Construction 8vo,  4  00 

Wegmann's  Construction  of  Masonry  Dams 4to,  5  00 

Wellington's  Location  of  Railways. Small  8vo,  5  00 

Wheeler's  Civil  Engineering 8vo,  4  00 

Wilson's  Topographical  Surveying 8vo,  3  50 

Wolff's  Windmill  as  a  Prime  Mover 8vo,  3  00 

HYDRAULICS. 

(See  also  ENGINEEKING,  p.  7.) 
Bazin's  Experiments  upon  the  Contraction  of  the  Liquid  Vein. 

(Trautwiue.) 8vo,  2  00 

Bovey 's  Treatise  on  Hydraulics 8vo,  4  00 

Church's  Mechanics  of  Engineering,  Solids,  and  Fluids. . .  .8vo,  6  00 

Coffin's  Graphical  Solution  of  Hydraulic  Problems 12mo,  2  50 

Fen-el's  Treatise  on  the  Winds,  Cyclones,  and  Tornadoes. .  .8vo,  4  00 

Folwel'i's  Water  Supply  Engineering 8vo,  4  00 

Frizell's  Water-power 8vo,  5  00 

Fuertes  s  Water  and  Public  Health 12mo,  1  50 

Water  Filtration  Works 12ino?  2  50 

Ganguillet  &  Kutter's  Flow  of  Water.     (Hering  &  Trautwine.) 

8vo,  4  00 

Hazeu's  Filtration  of  Public  Water  Supply 8vo,  3  00 

Hazlehurst  s  Towers  and  Tanks  for  Cities  aud  Towns 8vo,  2  50 

Herschel's  115  Experiments  8vo,  2  00 

Kiersted  s  Sewage  Disposal 12mo,  1  25 

Mason  s  Water  Supply 8vo.  5  00 

"     Examination  of  Water , 12mo,  1  25 

Merrimau's  Treatise  on  Hydraulics. 8vo,  4  00 

9 


Kichols's  Water  Supply  (Chemical  aud  Sanitary) 8vo,  $2  50 

Schuyler's  Reservoirs  for  Irrigation Large  8vo,  5  00 

Turneaure  and  Russell's  Public  Water-supplies 8vo,  5  00 

Wegmaun's  Water  Supply  of  the  City  of  New  York 4to,  10  00 

Weisbach's  Hydraulics.     (Du  Bois.) 8vo,  5  00 

Whipple's  Microscopy  of  Drinking  Water 8vo,  3  50 

Wilson's  Irrigation  Engineering ...  .8vo,  4  00 

"        Hydraulic  and  Placer  Mining 12mo,  2  00 

Wolff's  Windmill  as  a  Prime  Mover 8vo,  3  00 

Wood's  Theory  of  Turbines 8vo,  2  50 

LAW. 

Davis's  Elements  of  Law 8vo,  2  50 

' '      Treatise  on  Military  Law 8vo,  7  00 

Sheep,  7  50 

Manual  for  Courts-martial 16mo,  morocco,  1  50 

Wait's  Engineering  and  Architectural  Jurisprudence 8vo,  6  00 

Sheep,  6  50 

"      Law  of  Contracts 8vo,  300 

"      Law  of  Operations  Preliminary  to  Construction  in  En- 
gineering and  Architecture 8vo,  5  00 

Sheep,  5  50 

Winthrop's  Abridgment  of  Military  Law 12mo,  2  50 

MANUFACTURES. 

Allen's  Tables  for  Iron  Analysis 8vo,  3  00 

Beaumont's  Woollen  and  Worsted  Manufacture 12mo,  1  50 

Bolland's  Encyclopaedia  of  Founding  Terms 12mo.  3  00 

"         The  Iron  Founder 12mo,  250 

Supplement 12mo,  250 

Eissler's  Explosives,  Nitroglycerine  and  Dynamite 8vo,  4  00 

Ford  s  Boiler  Making  for  Boiler  Makers 18mo,  1  00 

Metcalfe's  Cost  of  Manufactures 8vo,  5  00 

Metcalf 's  Steel— A  Manual  for  Steel  Users 12mo,  2  00 

*  Reisig's  Guide  to  Piece  Dyeing 8vo,  25  00 

Spencer's  Sugar  Manufacturer's  Handbook  . . .  .16rno,  morocco,  2  00 
Handbook    for    Chemists    of    Beet    Sugar    Houses. 

16mo,  morocco,  3  00 

Thurston's  Manual  of  Steam  Boilers 8vo,  5  00 

Walke's  Lectures  on  Explosives 8vo,  4  00 

W  est's  American  Foundry  Practice 12mo,  2  50 

Moulder's  Text  book  12mo.  2  50 

Wiechmaun's  Sugar  Analysis Small  8vo,  2  50 

Woodbury's  Fire  Protection  of  Mills 8vo,  2  50 

10 


MATERIALS  OF  ENGINEERING. 

(See  also  ENGINEERING,  p.  7.) 

Baker's  Masonry  Construction 8vo,  $5  00 

Bovey's  Strength  of  Materials 8vo,  7  50 

Burr's  Elasticity  and  Resistance  of  Materials 8vo,  5  00 

By  rue's  Highway  Construction 8vo,  5  00 

Church's  Mechanics  of  Engineering — Solids  and  Fluids 8vo,  6  00 

Du  Bois's  Stresses  iu  Framed  Structures Siiiull  4to,  10  00 

Johnson's  Materials  of  Construction 8vo,  6  00 

Lanza's  Applied  Mechanics 3vo,  7  50 

Marteus's  Testing  Materials.     (Heuning.) 2  vols.,  8vo,  7  50 

Merrill's  Stones  for  Building  and  Decoration 8vo,  5  00 

Merriman's  Mechanics  of  Materials 8vo,  4  00 

"           Strength  of  Materials 12mo,  1  00 

Pattou's  Treatise  on  Foundations 8vo,  5  00 

Rockwell's  Roads  and  Pavements  in  France 12mo,  1  25 

Spalding's  Roads  and  Pavements 12mo,  2  00 

Thurstou's  Materials  of  Construction , 8vo,  5  00 

Materials  of  Engineering 3  vols. ,  8vo,  8  00 

Vol.  I. ,  Non-metallic  8vo,  2  00 

Vol.  II.,  Iron  and  Steel 8vo,  3  50 

Vol.  III.,  Alloys,  Brasses,  and  Bronzes 8vo,  2  50 

Wood's  Resistance  of  Materials 8vo,  2  00 

MATHEMATICS. 

Baker's  Elliptic  Functions 8vo,  1  50 

*Bass's  Differential  Calculus 12mo,  4  00 

Briggs's  Plane  Analytical  Geometry 12mo,  1  00 

Chapman's  Theory  of  Equations 12mo,  1  50 

Comptou's  Logarithmic  Computations 12mo,  1  50 

Davis's  Introduction  to  the  Logic  of  Algebra 8vo,  1  50 

Halsted's  Elements  of  Geometry Svo,  1  75 

"        Synthetic  Geometry 8vo,  1  50 

Johnson's  Curve  Tracing 12mo,  1  00 

"         Differential  Equations — Ordinary  and  Partial. 

Small  Svo,  3  50 

"        Integral  Calculus 12mo,  150 

"  "        Unabridged.     Small  Svo.    (In  press.) 

"        Least  Squares , 12mo,  1  50 

*Ludlow's  Logarithmic  and  Other  Tables.     (Bass.) Svo,  2  00 

*      "        Trigonometry  with  Tables.     (Bass.) Svo,  300 

*Mahan's  Descriptive  Geometry  (Stone  Cutting)  Svo,  1  50 

Merrimaii  and  Woodward's  Higher  Mathematics. Svo,  5  00 

11 


Merriinan's  Method  of  Least  Squares 8vo,  $2  00 

Rice  and  Johnson's  Differential  and  Integral  Calculus, 

2  vols.  in  1,  small  8vo,  2  50 

"                  Differential  Calculus Small  8vo,  3  00 

"  Abridgment  of  Differential  Calculus. 

Small  8vo,  1  50 

Totten's  Metrology 8vo,  2  50 

Warren's  Descriptive  Geometry 2  vols.,  8vo,  3  50 

"         Drafting  Instruments .12mo,  1  25 

"        Free-hand  Drawing 12mo,  100 

"        Linear  Perspective 12mo,  100 

"        Primary  Geometry 12mo,  75 

Plane  Problems 12mo,  1  25 

"        Problems  and  Theorems 8vo,  2  50 

"        Projection  Drawing 12mo,  1  50 

Wood's  Co-ordinate  Geometry 8vo,  2  00 

"       Trigonometry 12mo,  1  00 

Woolf's  Descriptive  Geometry Large  8vo,  3  00 

MECHANICS-MACHINERY. 

(See  also  ENGINEERING,  p.  7.) 

Baldwin's  Steam  Heating  for  Buildings .12mo,  2  50 

Barr's  Kinematics  of  Machinery 8vo,  2  50 

Benjamin's  Wrinkles  and  Eecipes 12ino,  2  00 

Chordal's  Letters  to  Mechanics 12mo,  2  00 

Church's  Mechanics  of  Engineering 8vo,  6  00 

"        Notes  and  Examples  in  Mechanics 8vo,  2  00 

Crehore's  Mechanics  of  the  Girder 8vo,  5  00 

Cromwell's  Belts  and  Pulleys 12mo,  1  50 

Toothed  Gearing 12mo,  1  50 

Compton's  First  Lessons  in  Metal  Working 12mo,  1  50 

Compton  and  De  Groodt's  Speed  Lathe 12mo,  1  50 

Dana's  Elementary  Mechanics 12mo,  1  50 

Dingey's  Machinery  Pattern  Making .12mo,  2  00 

*  Dredge's     Trans.     Exhibits    Building,     World     Exposition. 

Large  4to,  half  morocco,  5  00 

Du  Bois's  Mechanics.     Yol.  I.,  Kinematics „ 8vo,  3  50 

"                "               Vol.  II.,  Statics 8vo,  400 

"                "               Vol.  III.,  Kinetics 8vo,  350 

Fitzgerald's  Boston  Machinist 18mo,  1  00 

Flather's  Dynamometers 12mo,  2  00 

Rope  Driving 12mo,  200 

Hall's  Car  Lubrication 12mo,  1  00 

HoMy's  Saw  Filing 18mo,  75 

12 


*  Johnson's  Theoretical  Mechanics.     An  Elementary  Treatise. 

12mo,  $3  00 

Jones's  Machine  Design.     Part  I.,  Kinematics 8vo,  1  50 

"  Part  II.,  Strength  and  Proportion  of 

Machine  Parts 8vo,  3  00 

Lanza's  Applied  Mechanics 8vo,  7  50 

MacCord's  Kinematics 8vo,  5  00 

Merrimau's  Mechanics  of  Materials 8vo,  4  00 

Metcalfe's  Cost  of  Manufactures 8vo,  5  00 

*Michie's  Analytical  Mechanics 8vo,  4  00 

Richards's  Compressed  Air 12mo,  1  50 

Robinson's  Principles  of  Mechanism 8vo,  3  00 

Smith's  Press-working  of  Metals 8vo,  H  00 

Thurston's  Friction  and  Lost  Work 8vo,  3  00 

The  Animal  as  a  Machine 12mo,  1  00 

Warren's  Machine  Construction 2  vols.,  8vo,  7  50 

Weisbaclrs  Hydraulics  and  Hydraulic  Motors.    (Du  Bois.)..8vo,  5  00 
Mechanics    of    Engineering.      Vol.    III.,    Part   I., 

Sec.  I.     (Klein.) 8vo,  500 

Weisbach's   Mechanics    of  Engineering.     Vol.   III.,    Part   I., 

Sec.  II.     (Klein.) 8vo,  5  00 

Weisbach's  Steam  Engines.     (Du  Bois.) 8vo,  500 

Wood's  Analytical  Mechanics 8vo,  3  00 

"      Elementary  Mechanics 12mo,  125 

"                                  "           Supplement  and  Key 12ino,  1  25 

METALLURGY. 

Allen's  Tables  for  Iron  Analysis 8vo,  3  00 

Egleston's  Gold  and  Mercury Large  8vo,  7  50 

Metallurgy  of  Silver Large  8vo,  7  50 

*  Kerl's  Metallurgy— Steel,  Fuel,  etc 8vo,  15  00 

Kunhardt's  Ore  Dressing  in  Europe 8vo,  1  50 

Metcalf's  Steel— A  Manual  for  Steel  Users 12mo,  2  00 

O'Driscoll's  Treatment  of  Gold  Ores 8vo,  2  00 

Thurston's  Iron  and  Steel 8vo,  3  50 

Alloys 8vo,  250 

Wilson's  Cyanide  Processes '. 12mo,  1  50 

MINERALOGY   AND  MINING. 

Barringer's  Minerals  of  Commercial  Value Oblong  morocco,  2  50 

Beard's  Ventilation  of  Mines 12mo,  2  50 

Boyd's  Resources  of  South  Western  Virginia 8vo,  3  00 

Map  of  South  Western  Virginia Pocket-book  form,  2  00 

Brush  and  Penfield's  Determinative  Mineralogy.   New  Ed.  8vo,  4  00 

13 


Chester's  Catalogue  of  Minerals 8vo,  $1  25 

Paper,  50 

"       Dictionary  of  the  Names  of  Minerals .8vo,  3  00 

Dana's  American  Localities  of  Minerals Large  8vo,  1  00 

"      Descriptive  Mineralogy.  (E.S.)  Large  8vo.  half  morocco,  12  50 

"      First  Appendix  to  System  of  Mineralogy.   . .  .Large  8vo,  1  00 

"      Mineralogy  and  Petrography.     (J.  D.) 12mo,  2  00 

"      Minerals  and  How  to  Study  Them.     (E.  S.).. 12mo,  1  50 

"      Text-book  of  Mineralogy.     (E.  S.)..  .New  Edition.     8vo,  400 

*  Drinker's  Tunnelling,  Explosives,  Compounds,  and  Rock  Drills. 

4to,  half  morocco,  25  00 

Egleston's  Catalogue  of  Minerals  and  Synonyms 8vo,  2  50 

Eissler's  Explosives — Nitroglycerine  and  Dynamite 8vo,  4  00 

Hussak's  Rock  forming  Minerals.     (Smith.) Small  8vo,  2  00 

Ihlseng's  Manual  of  Mining . . 8vo,  4  00 

Kunhardt's  Ore  Dressing  in  Europe 8vo,  1  50 

O'Driscoll's  Treatment  of  Gold  Ores 8vo,  2  00 

*  Penfield's  Record  of  Mineral  Tests Paper,  8vo,  50 

Rosenbusch's    Microscopical    Physiography   of    Minerals    and 

Rocks.     (Idduigs.) 8vo,  500 

Sawyer's  Accidents  in  Mines Large  8vo,  7  00 

Stockbridge's  Rocks  and  Soils 8vo,  2  50 

*Tillman's  Important  Minerals  and  Rocks 8vo,  2  00 

"Walke's  Lectures  on  Explosives 8vo,  4  00 

Williams's  Lithology 8vo,  3  00 

Wilson's  Mine  Ventilation 12mo,  125 

Hydraulic  and  Placer  Mining 0 ......  12mo,  2  50 

STEAM  AND  ELECTRICAL  ENGINES,  BOILERS,  Etc. 

(See  also  ENGINEERING,  p.  7.) 

Baldwin's  Steam  Heating  for  Buildings 12mo»  2  50 

Clerk's  Gas  Engine Small  8vo,  4  00 

Ford's  Boiler  Making  for  Boiler  Makers 18mo,  1  00 

Hemen way's  Indicator  Practice 12mo,  2  00 

Kent's  Steam-boiler  Economy , .'. 8vo,  4  00 

Kneass's  Practice  and  Theory  of  the  Injector 8vo,  1  50 

MacCord's  Slide  Valve 8vo,  2  00 

Meyer's  Modern  Locomotive  Construction 4to,  10  00 

Peabody  and  Miller's  Steam-boilers 8vo,  4  00 

Peabody's  Tables  of  Saturated  Steam 8vo,  1  00 

"  Thermodynamics  of  the  Steam  Engine 8vo,  5  00 

Valve  Gears  for  the  Steam  Engine 8vo,  250 

"  Manual  of  the  Steam-engine  Indicator 12mo,  1  50 

Fray's  Twenty  Years  with  the  Indicator , , ,  .Large  8vo,  2  50 

14 


Pupin  and  Ostcrberg's  Thermodynamics 12mo,  $1  25 

Reagan's  Steam  and  Electric  Locomotives .  .12mo,  2  00 

Rontgen's  Thermodynamics.     (Du  Bois. ) 8vo,  5  00 

Sinclair's  Locomotive  Running 12mo,  2  00 

Snow 's  Steam-boiler  Practice 8vo.  3  00 

Thurston's  Boiler  Explosions 12mo,  1  50 

Engine  and  Boiler  Trials 8vo,  500 

"  Manual  of  the  Steam  Engine.      Part  I.,  Structure 

and  Theory 8vo,  6  00 

Manual  of   the    Steam  Engine.      Part  II.,  Design, 

Construction,  and  Operation 8vo,  6  00 

2  parts,  10  00 

"           Philosophy  of  the  Steam  Engine 12mo,  75 

"  Reflection,  on  the  Motive  Power  of  Heat.    (Caruot.) 

12mo,  1  50 

Stationary  Steam  Engines  8vo,  2  50 

"           Steam-boiler  Construction  and  Operation 8vo,  5  00 

Spangler's  Valve  Gears 8vo,  2  50 

Notes  on  Thermodynamics 12mo,  1  00 

Weisbach's  Steam  Engine.     (Du  Bois.) 8vo;  500 

Whitham's  Steam-engine  Design..,  .-, 8vo,  5  00 

Wilson's  Steam  Boilers.     (Flather.)       12mo,  250 

Wood's  Thermodynamics,  Heat  Motors,  etc 8vo,  4  00 

TABLES,  WEIGHTS,  AND  MEASURES. 

Adrian ce's  Laboratory  Calculations 12nio,  1  25 

Allen's  Tables  for  Iron  Analysis , .  .8vo,  3  00 

Bixby's  Graphical  Computing  Tables Sheet,  25 

Cornptou's  Logarithms 12mo,  1  50 

Crandall's  Railway  and  Earthwork  Tables 8vo,  1  50 

Davis's  Elevation  and  Stadia  Tables Small  8vo,  1  00 

Fisher's  Table  of  Cubic  Yards Cardboard,  25 

Hudson's  Excavation  Tables.     Vol.  II 8vo,  1  00 

Johnson's  Stadia  and  Earthwork  Tables 8vo,  1  25 

Ludlow's  Logarithmic  and  Other  Tables.     (Bass.) 12mo,  2  00 

Totten's  Metrology 8vo,  2  50 

VENTILATION. 

Baldwin's  Steam  Heating 12rno,  2  50 

Beard's  Ventilation  of  Mines. 12mo,  2  50 

Carpenter's  Heating  and  Ventilating  of  Buildings 8vo,  3  00 

Gerhard's  Sanitary  House  Inspection 12mo,  1  00 

Wilson's  Mine  Ventilation 12mo,  I  25 

15 


MISCELLANEOUS  PUBLICATIONS. 

Alcott's  Gems,  Sentiment,  Language Gilt  edges,  $5  00 

Emmou's  Geological  Guide-book  of  the  Rocky  Mountains.  .8vo,  1  50 

Ferrel' s  Treatise  ou  the  Winds 8vo,  4  00 

Haines's  Addresses  Delivered  before  the  Am.  Ry.  Assn.  ..12mo,  2  50 

Mott's  The  Fallacy  of  the  Present  Theory  of  Sound.  .Sq.  16mo,  1  00 

Richards's  Cost  of  Living 12mo,  1  00 

Ricketts's  History  of  Rensselaer  Polytechnic  Institute 8vo,  3  00 

Rotherham's    The    New    Testament    Critically    Emphasized. 

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Totten's  An  Important  Question  in  Metrology 8vo,  2  50 

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FOR  SCHOOLS  AND  THEOLOGICAL  SEMINARIES. 

Gesenius's  Hebrew  and   Chaldee  Lexicon  to  Old  Testament. 

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Green's  Elementary  Hebrew  Grammar 12mo,  1  25 

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"       Hebrew  Chrestomathy 8vo,  2  00 

Letteris's   Hebrew  Bible  (Massoretic  Notes  in  English). 

8vo,  arabesque,  2  25 

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Mott's  Composition,  Digestibility,  and  Nutritive  Value  of  Food. 

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Ruddiman's  Incompatibilities  in  Prescriptions 8vo,  2  00 

Steel's  Treatise  on  the  Diseases  of  the  Dog 8vo,  3  50 

WoodhulPs  Military  Hygiene 16mo,  1  50 

Worcester's  Small  Hospitals — Establishment  and  Maintenance, 
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